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Integer Exponents and Scientific Notation

Use the definition of a negative exponent

Use the Definition of a Negative Exponent

We saw that the Quotient Property for Exponents introduced earlier in this chapter, has two forms depending on whether the exponent is larger in the numerator or the denominator.

What if we just subtract exponents regardless of which is larger?

Let’s consider \(\frac{{x}^{2}}{{x}^{5}}\).

We subtract the exponent in the denominator from the exponent in the numerator.

\[\begin{array}{l}\frac{{x}^{2}}{{x}^{5}} \\ {x}^{2-5} \\ {x}^{-3}\end{array}\]

We can also simplify \(\frac{{x}^{2}}{{x}^{5}}\) by dividing out common factors:

This implies that \({x}^{-3}=\frac{1}{{x}^{3}}\) and it leads us to the definition of a negative exponent.

The negative exponent tells us we can re-write the expression by taking the reciprocal of the base and then changing the sign of the exponent.

Example

Try it.

Simplify: ⓐ \({4}^{-2}\) ⓑ \({10}^{-3}.\)

Solution
ⓐ\({4}^{-2}\)
Use the definition of a negative exponent, \({a}^{\text{-}n}=\frac{1}{{a}^{n}}\).\(\frac{1}{{4}^{2}}\)
Simplify.\(\frac{1}{16}\)
ⓑ\({10}^{-3}\)
Use the definition of a negative exponent, \({a}^{\text{-}n}=\frac{1}{{a}^{n}}\).\(\frac{1}{{10}^{3}}\)
Simplify.\(\frac{1}{1000}\)
\(\frac{1}{{a}^{\text{-}n}}\)
Use the definition of a negative exponent, \({a}^{\text{-}n}=\frac{1}{{a}^{n}}\).\(\frac{1}{\frac{1}{{a}^{n}}}\)
Simplify the complex fraction.\(1\cdot \frac{{a}^{n}}{1}\)
Multiply.\({a}^{n}\)
Example

Try it.

Simplify: ⓐ \(\frac{1}{{y}^{-4}}\) ⓑ \(\frac{1}{{3}^{-2}}.\)

Solution
ⓐ\(\frac{1}{{y}^{-4}}\)
Use the property of a negative exponent, \(\frac{1}{{a}^{\text{-}n}}={a}^{n}\).\({y}^{4}\)
ⓑ\(\frac{1}{{3}^{-2}}\)
Use the property of a negative exponent, \(\frac{1}{{a}^{\text{-}n}}={a}^{n}\).\({3}^{2}\)
Simplify.\(9\)

Condensed: the full section is in OpenStax Elementary Algebra 2e.

Simplify Expressions with Integer Exponents

All of the exponent properties we developed earlier in the chapter with whole number exponents apply to integer exponents, too. We restate them here for reference.

Example

Try it.

Simplify: ⓐ \({x}^{-4}\cdot {x}^{6}\) ⓑ \({y}^{-6}\cdot {y}^{4}\) ⓒ \({z}^{-5}\cdot {z}^{-3}.\)

Solution

ⓐ

\({x}^{-4}\cdot {x}^{6}\)
Use the Product Property, \({a}^{m}\cdot {a}^{n}={a}^{m+n}.\)\({x}^{-4+6}\)
Simplify.\({x}^{2}\)
ⓑ
\({y}^{-6}\cdot {y}^{4}\)
Notice the same bases, so add the exponents.\({y}^{-6+4}\)
Simplify.\({y}^{-2}\)
Use the definition of a negative exponent, \({a}^{\text{-}n}=\frac{1}{{a}^{n}}.\)\(\frac{1}{{y}^{2}}\)
ⓒ
\({z}^{-5}\cdot {z}^{-3}\)
Add the exponents, since the bases are the same.\({z}^{-5-3}\)
Simplify.\({z}^{-8}\)
Take the reciprocal and change the sign of the exponent,
using the definition of a negative exponent.		\(\frac{1}{{z}^{8}}\)

In the next two examples, we’ll start by using the Commutative Property to group the same variables together. This makes it easier to identify the like bases before using the Product Property.

Example

Try it.

Simplify: \(({m}^{4}{n}^{-3})({m}^{-5}{n}^{-2}).\)

Solution

\(({m}^{4}{n}^{-3})({m}^{-5}{n}^{-2})\)
Use the Commutative Property to get like bases together.\({m}^{4}{m}^{-5}\cdot {n}^{-2}{n}^{-3}\)
Add the exponents for each base.\({m}^{-1}\cdot {n}^{-5}\)
Take reciprocals and change the signs of the exponents.\(\frac{1}{{m}^{1}}\cdot \frac{1}{{n}^{5}}\)
Simplify.\(\frac{1}{m{n}^{5}}\)

Condensed: the full section is in OpenStax Elementary Algebra 2e.

Convert from Decimal Notation to Scientific Notation

Remember working with place value for whole numbers and decimals? Our number system is based on powers of 10. We use tens, hundreds, thousands, and so on. Our decimal numbers are also based on powers of tens, tenths, hundredths, thousandths, and so on. Consider the numbers 4,000 and \(0.004\). We know that 4,000 means \(4\ \times \ 1,000\) and 0.004 means \(4\ \times \ \frac{1}{1,000}\).

If we write the 1000 as a power of ten in exponential form, we can rewrite these numbers in this way:

\[\begin{array}{llll}4,000 & & & \ 0.004 \\ 4\ \times \ 1,000 & & & \ 4\ \times \ \frac{1}{1,000} \\ 4\ \times \ {10}^{3} & & & \ 4\ \times \ \frac{1}{{10}^{3}} \\ & & & \ 4\ \times \ {10}^{-3}\end{array}\]

When a number is written as a product of two numbers, where the first factor is a number greater than or equal to one but less than 10, and the second factor is a power of 10 written in exponential form, it is said to be in scientific notation.

It is customary in scientific notation to use as the \(\ \times\) multiplication sign, even though we avoid using this sign elsewhere in algebra.

If we look at what happened to the decimal point, we can see a method to easily convert from decimal notation to scientific notation.

In both cases, the decimal was moved 3 places to get the first factor between 1 and 10.

\(\begin{array}{llll}\text{The power of 10 is positive when the number is larger than 1:} & & & 4,000=4\ \times \ {10}^{3} \\ \text{The power of 10 is negative when the number is between 0 and 1:} & & & 0.004=4\ \times \ {10}^{-3}\end{array}\)

How to Convert from Decimal Notation to Scientific Notation

Try it.

Write in scientific notation: 37,000.

Solution

Condensed: the full section is in OpenStax Elementary Algebra 2e.

Convert Scientific Notation to Decimal Form

How can we convert from scientific notation to decimal form? Let’s look at two numbers written in scientific notation and

\[\begin{array}{llll}9.12\ \times \ {10}^{4} & & & \ 9.12\ \times \ {10}^{-4} \\ 9.12\ \times \ 10,000 & & & \ 9.12\ \times \ 0.0001 \\ 91,200 & & & \ 0.000912\end{array}\]

If we look at the location of the decimal point, we can see an easy method to convert a number from scientific notation to decimal form.

\[\begin{array}{llll}9.12\ \times \ {10}^{4}=91,200 & & & \ 9.12\ \times \ {10}^{-4}=0.000912\end{array}\]

In both cases the decimal point moved 4 places. When the exponent was positive, the decimal moved to the right. When the exponent was negative, the decimal point moved to the left.

How to Convert Scientific Notation to Decimal Form

Try it.

Convert to decimal form: \(6.2\ \times \ {10}^{3}.\)

Solution

The steps are summarized below.

Example

Try it.

Convert to decimal form: \(8.9\ \times \ {10}^{-2}.\)

Solution
Determine the exponent, n, on the factor 10.
Since the exponent is negative, move the decimal point 2 places to the left.
Add zeros as needed for placeholders.

Multiply and Divide Using Scientific Notation

Astronomers use very large numbers to describe distances in the universe and ages of stars and planets. Chemists use very small numbers to describe the size of an atom or the charge on an electron. When scientists perform calculations with very large or very small numbers, they use scientific notation. Scientific notation provides a way for the calculations to be done without writing a lot of zeros. We will see how the Properties of Exponents are used to multiply and divide numbers in scientific notation.

Example

Try it.

Multiply. Write answers in decimal form: \((4\ \times \ {10}^{5})(2\ \times \ {10}^{-7}).\)

Solution

\(\begin{array}{llll} & & & (4\ \times \ {10}^{5})(2\ \times \ {10}^{-7}) \\ \\ \\ \text{Use the Commutative Property to rearrange the factors.} & & & 4\cdot 2\cdot {10}^{5}\cdot {10}^{-7} \\ \\ \\ \text{Multiply.} & & & 8\ \times \ {10}^{-2} \\ \\ \\ \text{Change to decimal form by moving the decimal two places left.} & & & 0.08\end{array}\)

Example

Try it.

Divide. Write answers in decimal form: \(\frac{9\ \times \ {10}^{3}}{3\ \times \ {10}^{-2}}.\)

Solution

\(\frac{9\ \times \ {10}^{3}}{3\ \times \ {10}^{-2}}\)
Separate the factors, rewriting as the product of two fractions.\(\frac{9}{3}\ \times \ \frac{{10}^{3}}{{10}^{-2}}\)
Divide.\(3\ \times \ {10}^{5}\)
Change to decimal form by moving the decimal five places right.300,000

Key Concepts

  • Property of Negative Exponents
    • If \(n\) is a positive integer and \(a\ne 0\), then \(\frac{1}{{a}^{\text{-}n}}={a}^{n}\)
  • Quotient to a Negative Exponent
    • If \(a,b\) are real numbers, \(b\ne 0\) and \(n\) is an integer , then \({(\frac{a}{b})}^{\text{-}n}={(\frac{b}{a})}^{n}\)
  • To convert a decimal to scientific notation:
    1. Move the decimal point so that the first factor is greater than or equal to 1 but less than 10.
    2. Count the number of decimal places, \(n\), that the decimal point was moved.
    3. Write the number as a product with a power of 10. If the original number is:
      • greater than 1, the power of 10 will be \({10}^{n}\)
      • between 0 and 1, the power of 10 will be \({10}^{\text{-}n}\)
    4. Check.
  • To convert scientific notation to decimal form:
    1. Determine the exponent, \(n\), on the factor 10.
    2. Move the decimal \(n\)places, adding zeros if needed.
      • If the exponent is positive, move the decimal point \(n\) places to the right.
      • If the exponent is negative, move the decimal point \(|n|\) places to the left.
    3. Check.

Chapter 6 Review Exercises

Identify Polynomials, Monomials, Binomials and Trinomials

In the following exercises, determine if each of the following polynomials is a monomial, binomial, trinomial, or other polynomial.

Try it.

ⓐ \(11{c}^{4}-23{c}^{2}+1\)
ⓑ \(9{p}^{3}+6{p}^{2}-p-5\)
ⓒ \(\frac{3}{7}x+\frac{5}{14}\)
ⓓ 10
ⓔ \(2y-12\)

Try it.

ⓐ \({a}^{2}-{b}^{2}\)
ⓑ \(24{d}^{3}\)
ⓒ \({x}^{2}+8x-10\)
ⓓ \({m}^{2}{n}^{2}-2mn+6\)
ⓔ \(7{y}^{3}+{y}^{2}-2y-4\)

Solution

ⓐ binomial ⓑ monomial ⓒ trinomial ⓓ trinomial ⓔ other polynomial

Determine the Degree of Polynomials

In the following exercises, determine the degree of each polynomial.

Try it.

  1. ⓐ \(3{x}^{2}+9x+10\)
  2. ⓑ \(14{a}^{2}bc\)
  3. ⓒ \(6y+1\)
  4. ⓓ \({n}^{3}-4{n}^{2}+2n-8\)
  5. ⓔ \(-19\)

Try it.

  1. ⓐ \(5{p}^{3}-8{p}^{2}+10p-4\)
  2. ⓑ \(-20{q}^{4}\)
  3. ⓒ \({x}^{2}+6x+12\)
  4. ⓓ \(23{r}^{2}{s}^{2}-4rs+5\)
  5. ⓔ 100
Solution

ⓐ 3 ⓑ 4 ⓒ 2 ⓓ 4 ⓔ 0

Add and Subtract Monomials

In the following exercises, add or subtract the monomials.

Try it.

\({\ \text{5y}}^{\text{3}}+8{y}^{3}\)

Try it.

\(-14k+19k\)

Solution

\(5k\)

Try it.

\(12q-(-6q)\)

Try it.

\(\text{12x}-4y-9x\)

Try it.

\(3{m}^{2}+7{n}^{2}-3{m}^{2}\)

Solution

\(7{n}^{2}\)

Try it.

\(6{x}^{2}y-4x+8x{y}^{2}\)

Try it.

\(\text{13a}+b\)

Solution

\(\text{13a}+b\)

Add and Subtract Polynomials

In the following exercises, add or subtract the polynomials.

Try it.

\((5{x}^{2}+12x+1)+(6{x}^{2}-8x+3)\)

Try it.

\((9{p}^{2}-5p+3)+(4{p}^{2}-4)\)

Solution

\(13{p}^{2}-5p-1\)

Try it.

\((10{m}^{2}-8m-1)-(5{m}^{2}+m-2)\)

Try it.

\((7{y}^{2}-8y)-(y-4)\)

Solution

\(7{y}^{2}-9y+4\)

Try it.

Subtract
\((3{s}^{2}+10)\ \text{from}\ (15{s}^{2}-2s+8)\)

Try it.

Find the sum of \(({a}^{2}+6a+9)\ \text{and}\ (5{a}^{3}-7)\)

Solution

\(5{a}^{3}+{a}^{2}+6a+2\)

Evaluate a Polynomial for a Given Value of the Variable

In the following exercises, evaluate each polynomial for the given value.

Try it.

Evaluate \(3{y}^{2}-y+1\) when:

  1. ⓐ \(y=5\)
  2. ⓑ \(y=-1\)
  3. ⓒ \(y=0\)

Try it.

Evaluate \(10-12x\) when:

  1. ⓐ \(x=3\)
  2. ⓑ \(x=0\)
  3. ⓒ \(x=-1\)
Solution

ⓐ \(-26\) ⓑ 10 ⓒ 22

Try it.

Randee drops a stone off the 200 foot high cliff into the ocean. The polynomial \(-16{t}^{2}+200\) gives the height of a stone \(t\) seconds after it is dropped from the cliff. Find the height after \(t=3\) seconds.

Try it.

A manufacturer of stereo sound speakers has found that the revenue received from selling the speakers at a cost of p dollars each is given by the polynomial \(-4{p}^{2}+460p.\) Find the revenue received when \(p=75\) dollars.

Solution

12,000

Condensed: the full section is in OpenStax Elementary Algebra 2e.

Chapter Practice Test

In the following exercises, simplify each expression.

In the following exercises, simplify, and write your answer in decimal form.

Use the Definition of a Negative Exponent

The Quotient Property of Exponents, introduced in Divide Monomials, had two forms depending on whether the exponent in the numerator or denominator was larger.

What if we just subtract exponents, regardless of which is larger? Let’s consider \(\frac{{x}^{2}}{{x}^{5}}.\)

We subtract the exponent in the denominator from the exponent in the numerator.

\[\frac{{x}^{2}}{{x}^{5}}\]\[{x}^{2-5}\]\[{x}^{-3}\]

We can also simplify \(\frac{{x}^{2}}{{x}^{5}}\) by dividing out common factors: \(\frac{{x}^{2}}{{x}^{5}}.\)

This implies that \({x}^{-3}=\frac{1}{{x}^{3}}\) and it leads us to the definition of a negative exponent.

The negative exponent tells us to re-write the expression by taking the reciprocal of the base and then changing the sign of the exponent. Any expression that has negative exponents is not considered to be in simplest form. We will use the definition of a negative exponent and other properties of exponents to write an expression with only positive exponents.

Example

Try it.

Simplify:

  1. ⓐ \(\ {4}^{-2}\)
  2. ⓑ \(\ {10}^{-3}\)

Solution
ⓐ
\({4}^{-2}\)
Use the definition of a negative exponent, \({a}^{-n}=\frac{1}{{a}^{n}}.\)\(\frac{1}{{4}^{2}}\)
Simplify.\(\frac{1}{16}\)
ⓑ
\({10}^{-3}\)
Use the definition of a negative exponent, \({a}^{-n}=\frac{1}{{a}^{n}}.\)\(\frac{1}{{10}^{3}}\)
Simplify.\(\frac{1}{1000}\)

When simplifying any expression with exponents, we must be careful to correctly identify the base that is raised to each exponent.

Example

Try it.

Simplify:

  1. ⓐ \(\ {(-3)}^{-2}\)
  2. ⓑ \(\ {-3}^{-2}\)

Solution

The negative in the exponent does not affect the sign of the base.

ⓐ
The exponent applies to the base, \(-3\).\({(-3)}^{-2}\)
Take the reciprocal of the base and change the sign of the exponent.\(\frac{1}{{(-3)}^{2}}\)
Simplify.\(\frac{1}{9}\)
ⓑ
The expression \(-{3}^{-2}\) means "find the opposite of \({3}^{-2}\)".
The exponent applies only to the base, 3.		\(-{3}^{-2}\)
Rewrite as a product with −1.\(-1\cdot {3}^{-2}\)
Take the reciprocal of the base and change the sign of the exponent.\(-1\cdot \frac{1}{{3}^{2}}\)
Simplify.\(-\frac{1}{9}\)

Condensed: the full section is in OpenStax Prealgebra 2e.

Simplify Expressions with Integer Exponents

All the exponent properties we developed earlier in this chapter with whole number exponents apply to integer exponents, too. We restate them here for reference.

Example

Try it.

Simplify:

  1. ⓐ \(\ {x}^{-4}\cdot {x}^{6}\)
  2. ⓑ \(\ {y}^{-6}\cdot {y}^{4}\)
  3. ⓒ \(\ {z}^{-5}\cdot {z}^{-3}\)

Solution
ⓐ
\({x}^{-4}\cdot {x}^{6}\)
Use the Product Property, \({a}^{m}\cdot {a}^{n}={a}^{m+n}.\)\({x}^{-4+6}\)
Simplify.\({x}^{2}\)
ⓑ
\({y}^{-6}\cdot {y}^{4}\)
The bases are the same, so add the exponents.\({y}^{-6+4}\)
Simplify.\({y}^{-2}\)
Use the definition of a negative exponent, \({a}^{-n}=\frac{1}{{a}^{n}}.\)\(\frac{1}{{y}^{2}}\)
ⓒ
\({z}^{-5}\cdot {z}^{-3}\)
The bases are the same, so add the exponents.\({z}^{-5-3}\)
Simplify.\({z}^{-8}\)
Use the definition of a negative exponent, \({a}^{-n}=\frac{1}{{a}^{n}}.\)\(\frac{1}{{z}^{8}}\)

In the next two examples, we’ll start by using the Commutative Property to group the same variables together. This makes it easier to identify the like bases before using the Product Property of Exponents.

Example

Try it.

Simplify: \(({m}^{4}{n}^{-3})({m}^{-5}{n}^{-2}).\)

Solution
\(({m}^{4}{n}^{-3})({m}^{-5}{n}^{-2})\)
Use the Commutative Property to get like bases together.\({m}^{4}{m}^{-5}\cdot {n}^{-2}{n}^{-3}\)
Add the exponents for each base.\({m}^{-1}\cdot {n}^{-5}\)
Take reciprocals and change the signs of the exponents.\(\frac{1}{{m}^{1}}\cdot \frac{1}{{n}^{5}}\)
Simplify.\(\frac{1}{m{n}^{5}}\)

If the monomials have numerical coefficients, we multiply the coefficients, just as we did in Use Multiplication Properties of Exponents.

Condensed: the full section is in OpenStax Prealgebra 2e.

Convert from Decimal Notation to Scientific Notation

Remember working with place value for whole numbers and decimals? Our number system is based on powers of \(10.\) We use tens, hundreds, thousands, and so on. Our decimal numbers are also based on powers of tens, tenths, hundredths, thousandths, and so on.

Consider the numbers \(4000\) and \(0.004.\) We know that \(4000\) means \(4\ \times \ 1000\) and \(0.004\) means \(4\ \times \ \frac{1}{1000}.\) If we write the \(1000\) as a power of ten in exponential form, we can rewrite these numbers in this way:

\[\begin{array}{llll}4000 & \ & & 0.004 \\ 4\ \times \ 1000 & \ & & 4\ \times \ \frac{1}{1000} \\ 4\ \times \ {10}^{3} & \ & & 4\ \times \ \frac{1}{{10}^{3}} \\ & \ & & 4\ \times \ {10}^{-3}\end{array}\]

When a number is written as a product of two numbers, where the first factor is a number greater than or equal to one but less than \(10,\) and the second factor is a power of \(10\) written in exponential form, it is said to be in scientific notation.

It is customary in scientific notation to use \(\ \times\) as the multiplication sign, even though we avoid using this sign elsewhere in algebra.

Scientific notation is a useful way of writing very large or very small numbers. It is used often in the sciences to make calculations easier.

If we look at what happened to the decimal point, we can see a method to easily convert from decimal notation to scientific notation.

In both cases, the decimal was moved \(3\) places to get the first factor, \(4,\) by itself.

  • The power of \(10\) is positive when the number is larger than \(1\text{:}\ 4000=4\ \times \ {10}^{3}.\)
  • The power of \(10\) is negative when the number is between \(0\) and \(1\text{:}\ 0.004=4\ \times \ {10}^{-3}.\)
Example

Try it.

Write \(37,000\) in scientific notation.

Solution
Step 1: Move the decimal point so that the first factor is greater than or equal to 1 but less than 10.
Step 2: Count the number of decimal places, \(n\), that the decimal point was moved.3.70000
4 places
Step 3: Write the number as a product with a power of 10.\(3.7\times {10}^{4}\)
If the original number is:
  • greater than 1, the power of 10 will be \({10}^{n}\).
  • between 0 and 1, the power of 10 will be \({10}^{-n}\)
Step 4: Check.
\({10}^{4}\) is 10,000 and 10,000 times 3.7 will be 37,000.
\(37,000=3.7\times {10}^{4}\)

Condensed: the full section is in OpenStax Prealgebra 2e.

Convert Scientific Notation to Decimal Form

How can we convert from scientific notation to decimal form? Let’s look at two numbers written in scientific notation and

\[\begin{array}{llll}9.12\ \times \ {10}^{4} & \ & & 9.12\ \times \ {10}^{-4} \\ 9.12\ \times \ 10,000 & \ & & 9.12\ \times \ 0.0001 \\ 91,200 & \ & & 0.000912\end{array}\]

If we look at the location of the decimal point, we can see an easy method to convert a number from scientific notation to decimal form.

In both cases the decimal point moved 4 places. When the exponent was positive, the decimal moved to the right. When the exponent was negative, the decimal point moved to the left.

Example

Try it.

Convert to decimal form: \(6.2\ \times \ {10}^{3}.\)

Solution
Step 1: Determine the exponent, \(n\), on the factor 10.\(6.2\times {10}^{3}\)
Step 2: Move the decimal point \(n\) places, adding zeros if needed.
  • If the exponent is positive, move the decimal point \(n\) places to the right.
  • If the exponent is negative, move the decimal point \(|n|\) places to the left.
6,200
Step 3: Check to see if your answer makes sense.
\({10}^{3}\) is 1000 and 1000 times 6.2 will be 6,200.\(6.2\times {10}^{3}=6,200\)
Example

Try it.

Convert to decimal form: \(8.9\ \times \ {10}^{-2}.\)

Solution
\(8.9\times {10}^{-2}\)
Determine the exponent \(n\), on the factor 10.The exponent is −2.
Move the decimal point 2 places to the left.
Add zeros as needed for placeholders.0.089
\(8.9\times {10}^{-2}=0.089\)
The Check is left to you.

Multiply and Divide Using Scientific Notation

We use the Properties of Exponents to multiply and divide numbers in scientific notation.

Example

Try it.

Multiply. Write answers in decimal form: \((4\ \times \ {10}^{5})(2\ \times \ {10}^{-7}).\)

Solution
\((4\ \times \ {10}^{5})(2\ \times \ {10}^{-7})\)
Use the Commutative Property to rearrange the factors.\(4\cdot 2\cdot {10}^{5}\cdot {10}^{-7}\)
Multiply 4 by 2 and use the Product Property to multiply \({10}^{5}\) by \({10}^{-7}\).\(8\ \times \ {10}^{-2}\)
Change to decimal form by moving the decimal two places left.\(0.08\)
Example

Try it.

Divide. Write answers in decimal form: \(\frac{9\ \times \ {10}^{3}}{3\ \times \ {10}^{-2}}.\)

Solution
\(\frac{9\ \times \ {10}^{3}}{3\ \times \ {10}^{-2}}\)
Separate the factors.\(\frac{9}{3}\ \times \ \frac{{10}^{3}}{{10}^{-2}}\)
Divide 9 by 3 and use the Quotient Property to divide \({10}^{3}\) by \({10}^{-2}\).\(3\ \times \ {10}^{5}\)
Change to decimal form by moving the decimal five places right.\(300,000\)

Key Concepts

  • Summary of Exponent Properties
    • If \(a,b\) are real numbers and \(m,n\) are integers, then \[\begin{array}{llll}\text{Product Property} & & & {a}^{m}\cdot {a}^{n}={a}^{m+n} \\ \text{Power Property} & & & {({a}^{m})}^{n}={a}^{m\cdot n} \\ \text{Product to a Power Property} & & & {(ab)}^{m}={a}^{m}{b}^{m} \\ \text{Quotient Property} & & & \frac{{a}^{m}}{{a}^{n}}={a}^{m-n},\ a\ne 0 \\ \text{Zero Exponent Property} & & & {a}^{0}=1,\ a\ne 0 \\ \text{Quotient to a Power Property} & & & {(\frac{a}{b})}^{m}=\frac{{a}^{m}}{{b}^{m}},\ b\ne 0 \\ \text{Definition of Negative Exponent} & & & {a}^{-n}=\frac{1}{{a}^{n}}\end{array}\]
  • Convert from Decimal Notation to Scientific Notation: To convert a decimal to scientific notation:
    1. Move the decimal point so that the first factor is greater than or equal to 1 but less than 10.
    2. Count the number of decimal places, \(n\), that the decimal point was moved.
    3. Write the number as a product with a power of 10.
      • If the original number is greater than 1, the power of 10 will be \({10}^{n}\).
      • If the original number is between 0 and 1, the power of 10 will be \({10}^{-n}\).
    4. Check.
  • Convert Scientific Notation to Decimal Form: To convert scientific notation to decimal form:
    1. Determine the exponent, \(n\), on the factor 10.
    2. Move the decimal \(n\) places, adding zeros if needed.
      • If the exponent is positive, move the decimal point \(n\) places to the right.
      • If the exponent is negative, move the decimal point \(|n|\) places to the left.
    3. Check.

Integer Exponents and Scientific Notation

Use the Definition of a Negative Exponent

In the following exercises, simplify.

Try it.

\({5}^{-3}\)

Try it.

\({8}^{-2}\)

Solution

\(\frac{1}{64}\)

Try it.

\({3}^{-4}\)

Try it.

\({2}^{-5}\)

Solution

\(\frac{1}{32}\)

Try it.

\({7}^{-1}\)

Try it.

\({10}^{-1}\)

Solution

\(\frac{1}{10}\)

Try it.

\({2}^{-3}+{2}^{-2}\)

Try it.

\({3}^{-2}+{3}^{-1}\)

Solution

\(\frac{4}{9}\)

Try it.

\({3}^{-1}+{4}^{-1}\)

Try it.

\({10}^{-1}+{2}^{-1}\)

Solution

\(\frac{3}{5}\)

Try it.

\({10}^{0}-{10}^{-1}+{10}^{-2}\)

Try it.

\({2}^{0}-{2}^{-1}+{2}^{-2}\)

Solution

\(\frac{3}{4}\)

Try it.

  1. ⓐ \(\ {(-6)}^{-2}\)
  2. ⓑ \(\ -{6}^{-2}\)

Try it.

  1. ⓐ \(\ {(-8)}^{-2}\)
  2. ⓑ \(\ -{8}^{-2}\)

Solution

  1. ⓐ \(\ \frac{1}{64}\)
  2. ⓑ \(\ -\frac{1}{64}\)

Try it.

  1. ⓐ \(\ {(-10)}^{-4}\)
  2. ⓑ \(\ -{10}^{-4}\)

Try it.

  1. ⓐ \(\ {(-4)}^{-6}\)
  2. ⓑ \(\ -{4}^{-6}\)

Solution

  1. ⓐ \(\ \frac{1}{4096}\)
  2. ⓑ \(\ -\frac{1}{4096}\)

Try it.

  1. ⓐ \(\ 5\cdot {2}^{-1}\)
  2. ⓑ \(\ {(5\cdot 2)}^{-1}\)

Try it.

  1. ⓐ \(\ 10\cdot {3}^{-1}\)
  2. ⓑ \(\ {(10\cdot 3)}^{-1}\)

Solution

  1. ⓐ \(\ \frac{10}{3}\)
  2. ⓑ \(\ \frac{1}{30}\)

Try it.

  1. ⓐ \(\ 4\cdot {10}^{-3}\)
  2. ⓑ \(\ {(4\cdot 10)}^{-3}\)

Try it.

  1. ⓐ \(\ 3\cdot {5}^{-2}\)
  2. ⓑ \(\ {(3\cdot 5)}^{-2}\)

Solution

  1. ⓐ \(\ \frac{3}{25}\)
  2. ⓑ \(\ \frac{1}{225}\)

Try it.

\({n}^{-4}\)

Try it.

\({p}^{-3}\)

Solution

\(\frac{1}{{p}^{3}}\)

Try it.

\({c}^{-10}\)

Try it.

\({m}^{-5}\)

Solution

\(\frac{1}{{m}^{5}}\)

Try it.

  1. ⓐ \(\ 4{x}^{-1}\)
  2. ⓑ \(\ {(4x)}^{-1}\)
  3. ⓒ \(\ {(-4x)}^{-1}\)

Try it.

  1. ⓐ \(\ 3{q}^{-1}\)
  2. ⓑ \(\ {(3q)}^{-1}\)
  3. ⓒ \(\ {(-3q)}^{-1}\)

Solution

  1. ⓐ \(\ \frac{3}{q}\)
  2. ⓑ \(\ \frac{1}{3q}\)
  3. ⓒ \(\ -\frac{1}{3q}\)

Try it.

  1. ⓐ \(\ 6{m}^{-1}\)
  2. ⓑ \(\ {(6m)}^{-1}\)
  3. ⓒ \(\ {(-6m)}^{-1}\)

Try it.

  1. ⓐ \(\ 10{k}^{-1}\)
  2. ⓑ \(\ {(10k)}^{-1}\)
  3. ⓒ \(\ {(-10k)}^{-1}\)

Solution

  1. ⓐ \(\ \frac{10}{k}\)
  2. ⓑ \(\ \frac{1}{10k}\)
  3. ⓒ \(\ -\frac{1}{10k}\)

Simplify Expressions with Integer Exponents

In the following exercises, simplify.

Try it.

\({p}^{-4}\cdot {p}^{8}\)

Try it.

\({r}^{-2}\cdot {r}^{5}\)

Solution

r3

Try it.

\({n}^{-10}\cdot {n}^{2}\)

Try it.

\({q}^{-8}\cdot {q}^{3}\)

Solution

\(\frac{1}{{q}^{5}}\)

Try it.

\({k}^{-3}\cdot {k}^{-2}\)

Try it.

\({z}^{-6}\cdot {z}^{-2}\)

Solution

\(\frac{1}{{z}^{8}}\)

Try it.

\(a\cdot {a}^{-4}\)

Try it.

\(m\cdot {m}^{-2}\)

Solution

\(\frac{1}{m}\)

Try it.

\({p}^{5}\cdot {p}^{-2}\cdot {p}^{-4}\)

Try it.

\({x}^{4}\cdot {x}^{-2}\cdot {x}^{-3}\)

Solution

\(\frac{1}{x}\)

Try it.

\({a}^{3}{b}^{-3}\)

Try it.

\({u}^{2}{v}^{-2}\)

Solution

\(\frac{{u}^{2}}{{v}^{2}}\)

Try it.

\(({x}^{5}{y}^{-1})({x}^{-10}{y}^{-3})\)

Try it.

\(({a}^{3}{b}^{-3})({a}^{-5}{b}^{-1})\)

Solution

\(\frac{1}{{a}^{2}{b}^{4}}\)

Try it.

\((u{v}^{-2})({u}^{-5}{v}^{-4})\)

Try it.

\((p{q}^{-4})({p}^{-6}{q}^{-3})\)

Solution

\(\frac{1}{{p}^{5}{q}^{7}}\)

Try it.

\((-2{r}^{-3}{s}^{9})(6{r}^{4}{s}^{-5})\)

Try it.

\((-3{p}^{-5}{q}^{8})(7{p}^{2}{q}^{-3})\)

Solution

\(-\frac{21{q}^{5}}{{p}^{3}}\)

Try it.

\((-6{m}^{-8}{n}^{-5})(-9{m}^{4}{n}^{2})\)

Try it.

\((-8{a}^{-5}{b}^{-4})(-4{a}^{2}{b}^{3})\)

Solution

\(\frac{32}{{a}^{3}b}\)

Try it.

\({({a}^{3})}^{-3}\)

Try it.

\({({q}^{10})}^{-10}\)

Solution

\(\frac{1}{{q}^{100}}\)

Try it.

\({({n}^{2})}^{-1}\)

Try it.

\({({x}^{4})}^{-1}\)

Solution

\(\frac{1}{{x}^{4}}\)

Try it.

\({({y}^{-5})}^{4}\)

Try it.

\({({p}^{-3})}^{2}\)

Solution

\(\frac{1}{{p}^{6}}\)

Try it.

\({({q}^{-5})}^{-2}\)

Try it.

\({({m}^{-2})}^{-3}\)

Solution

m6

Try it.

\({(4{y}^{-3})}^{2}\)

Try it.

\({(3{q}^{-5})}^{2}\)

Solution

\(\frac{9}{{q}^{10}}\)

Try it.

\({(10{p}^{-2})}^{-5}\)

Try it.

\({(2{n}^{-3})}^{-6}\)

Solution

\(\frac{{n}^{18}}{64}\)

Try it.

\(\frac{{u}^{9}}{{u}^{-2}}\)

Try it.

\(\frac{{b}^{5}}{{b}^{-3}}\)

Solution

b8

Try it.

\(\frac{{x}^{-6}}{{x}^{4}}\)

Try it.

\(\frac{{m}^{5}}{{m}^{-2}}\)

Solution

m7

Try it.

\(\frac{{q}^{3}}{{q}^{12}}\)

Try it.

\(\frac{{r}^{6}}{{r}^{9}}\)

Solution

\(\frac{1}{{r}^{3}}\)

Try it.

\(\frac{{n}^{-4}}{{n}^{-10}}\)

Try it.

\(\frac{{p}^{-3}}{{p}^{-6}}\)

Solution

p3

Convert from Decimal Notation to Scientific Notation

In the following exercises, write each number in scientific notation.

Try it.

8,750,000

Try it.

1,290,000

Solution

1.29 × 106

Try it.

0.00000924

Try it.

0.0000103

Solution

1.03 × 10−5

Try it.

The population of the United States on July 4, 2010 was almost \(310,000,000.\)

Try it.

The population of the world on July 4, 2010 was more than \(6,850,000,000.\)

Solution

6.85 × 109

Try it.

The average width of a human hair is \(0.0018\) centimeters.

Try it.

The probability of winning the \(2010\) Megamillions lottery is about \(0.0000000057.\)

Solution

5.7 × 10−9

Convert Scientific Notation to Decimal Form

In the following exercises, convert each number to decimal form.

Try it.

\(4.1\ \times \ {10}^{2}\)

Try it.

\(8.3\ \times \ {10}^{2}\)

Solution

830

Try it.

\(5.5\ \times \ {10}^{8}\)

Try it.

\(1.6\ \times \ {10}^{10}\)

Solution

16,000,000,000

Try it.

\(3.5\ \times \ {10}^{-2}\)

Try it.

\(2.8\ \times \ {10}^{-2}\)

Solution

0.028

Try it.

\(1.93\ \times \ {10}^{-5}\)

Try it.

\(6.15\ \times \ {10}^{-8}\)

Solution

0.0000000615

Try it.

In 2010, the number of Facebook users each day who changed their status to ‘engaged’ was \(2\ \times \ {10}^{4}.\)

Try it.

At the start of 2012, the US federal budget had a deficit of more than \(\text{\$1.5}\ \times \ {10}^{13}.\)

Solution

$15,000,000,000,000

Try it.

The concentration of carbon dioxide in the atmosphere is \(3.9\ \times \ {10}^{-4}.\)

Try it.

The width of a proton is \(1\ \times \ {10}^{-5}\) of the width of an atom.

Solution

0.00001

Multiply and Divide Using Scientific Notation

In the following exercises, multiply or divide and write your answer in decimal form.

Try it.

\((2\ \times \ {10}^{5})(2\ \times \ {10}^{-9})\)

Try it.

\((3\ \times \ {10}^{2})(1\ \times \ {10}^{-5})\)

Solution

0.003

Try it.

\((1.6\ \times \ {10}^{-2})(5.2\ \times \ {10}^{-6})\)

Try it.

\((2.1\ \times \ {10}^{-4})(3.5\ \times \ {10}^{-2})\)

Solution

0.00000735

Try it.

\(\frac{6\ \times \ {10}^{4}}{3\ \times \ {10}^{-2}}\)

Try it.

\(\frac{8\ \times \ {10}^{6}}{4\ \times \ {10}^{-1}}\)

Solution

20,000,000

Try it.

\(\frac{7\ \times \ {10}^{-2}}{1\ \times \ {10}^{-8}}\)

Try it.

\(\frac{5\ \times \ {10}^{-3}}{1\ \times \ {10}^{-10}}\)

Solution

50,000,000

Condensed: the full section is in OpenStax Prealgebra 2e.

Nou jy Geen sakrekenaar los hierdie een nie, maar die stukke daarvan is kon nie wees kondifiseerbaar. Probeer een hieronder, of tik jou eie.

Artikel (40)

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  1. What is the place value of the \(6\) in the number \(64,891\)?

    Openbaar die antwoord

    Ten thousands

  2. Name the decimal: \(0.0012.\)

    Openbaar die antwoord

    Twelve ten thousandths

  3. Subtract: \(5-(-3).\)

    Openbaar die antwoord

    \(8\)

  4. Simplify: ⓐ \({4}^{-2}\) ⓑ \({10}^{-3}.\)

    Openbaar die antwoord
    ⓐ\({4}^{-2}\)
    Use the definition of a negative exponent, \({a}^{\text{-}n}=\frac{1}{{a}^{n}}\).\(\frac{1}{{4}^{2}}\)
    Simplify.\(\frac{1}{16}\)
    ⓑ\({10}^{-3}\)
    Use the definition of a negative exponent, \({a}^{\text{-}n}=\frac{1}{{a}^{n}}\).\(\frac{1}{{10}^{3}}\)
    Simplify.\(\frac{1}{1000}\)

  5. Simplify: ⓐ \({2}^{-3}\) ⓑ \({10}^{-7}.\)

    Openbaar die antwoord

    ⓐ \(\frac{1}{8}\) ⓑ \(\frac{1}{{10}^{7}}\)

  6. Simplify: ⓐ \({3}^{-2}\) ⓑ \({10}^{-4}.\)

    Openbaar die antwoord

    ⓐ \(\frac{1}{9}\) ⓑ \(\frac{1}{10,000}\)

  7. Simplify: ⓐ \(\frac{1}{{y}^{-4}}\) ⓑ \(\frac{1}{{3}^{-2}}.\)

    Openbaar die antwoord
    ⓐ\(\frac{1}{{y}^{-4}}\)
    Use the property of a negative exponent, \(\frac{1}{{a}^{\text{-}n}}={a}^{n}\).\({y}^{4}\)
    ⓑ\(\frac{1}{{3}^{-2}}\)
    Use the property of a negative exponent, \(\frac{1}{{a}^{\text{-}n}}={a}^{n}\).\({3}^{2}\)
    Simplify.\(9\)

  8. Simplify: ⓐ \(\frac{1}{{p}^{-8}}\) ⓑ \(\frac{1}{{4}^{-3}}.\)

    Openbaar die antwoord

    ⓐ \({p}^{8}\) ⓑ \(64\)

  9. Simplify: ⓐ \(\frac{1}{{q}^{-7}}\) ⓑ \(\frac{1}{{2}^{-4}}.\)

    Openbaar die antwoord

    ⓐ \({q}^{7}\) ⓑ \(16\)

  10. Simplify: ⓐ \({(\frac{5}{7})}^{-2}\) ⓑ \({(-\frac{2x}{y})}^{-3}.\)

    Openbaar die antwoord
    ⓐ\({(\frac{5}{7})}^{-2}\)
    Use the Quotient to a Negative Exponent Property, \({(\frac{a}{b})}^{\text{-}n}={(\frac{b}{a})}^{n}\).
    Take the reciprocal of the fraction and change the sign of the exponent.\({(\frac{7}{5})}^{2}\)
    Simplify.\(\frac{49}{25}\)
    ⓑ\({(-\frac{2x}{y})}^{-3}\)
    Use the Quotient to a Negative Exponent Property, \({(\frac{a}{b})}^{\text{-}n}={(\frac{b}{a})}^{n}\).
    Take the reciprocal of the fraction and change the sign of the exponent.\({(-\frac{y}{2x})}^{3}\)
    Simplify.\(-\frac{{y}^{3}}{8{x}^{3}}\)

  11. Simplify: ⓐ \({(\frac{2}{3})}^{-4}\) ⓑ \({(-\frac{6m}{n})}^{-2}.\)

    Openbaar die antwoord

    ⓐ \(\frac{81}{16}\) ⓑ \(\frac{{n}^{2}}{36{m}^{2}}\)

  12. Simplify: ⓐ \({(\frac{3}{5})}^{-3}\) ⓑ \({(-\frac{a}{2b})}^{-4}.\)

    Openbaar die antwoord

    ⓐ \(\frac{125}{27}\) ⓑ \(\frac{16{b}^{4}}{{a}^{4}}\)

  13. Simplify: ⓐ \({(-3)}^{-2}\) ⓑ \(\text{-}{3}^{-2}\) ⓒ \({(-\frac{1}{3})}^{-2}\) ⓓ \(\text{-}{(\frac{1}{3})}^{-2}.\)

    Openbaar die antwoord
    ⓐ Here the exponent applies to the base \(-3\).\({(-3)}^{-2}\)
    Take the reciprocal of the base and change the sign of the exponent.\(\frac{1}{{(-3)}^{-2}}\)
    Simplify.\(\frac{1}{9}\)
    ⓑ The expression \(\text{-}{3}^{-2}\) means "find the opposite of \({3}^{-2}\)." Here the exponent applies to the base \({(-\frac{1}{3})}^{}\).\(\text{-}{3}^{-2}\)
    Rewrite as a product with \(-1\).\(-1\cdot {3}^{-2}\)
    Take the reciprocal of the base and change the sign of the exponent.\(-1\cdot \frac{1}{{3}^{2}}\)
    Simplify.\(-\frac{1}{9}\)
    ⓒ Here the exponent applies to the base \({(-\frac{1}{3})}^{}\).\({(-\frac{1}{3})}^{-2}\)
    Take the reciprocal of the base and change the sign of the exponent.\({(-\frac{3}{1})}^{2}\)
    Simplify.\(9\)
    ⓓ The expression \(\text{-}{(\frac{1}{3})}^{-2}\) means "find the opposite of \({(\frac{1}{3})}^{-2}\)." Here the exponent applies to the base \((\frac{1}{3})\).\(\)
    Rewrite as a product with \(-1\).\(-1\cdot {(\frac{1}{3})}^{-2}\)
    Take the reciprocal of the base and change the sign of the exponent.\(-1\cdot {(\frac{3}{1})}^{2}\)
    Simplify.\(-9\)

  14. Simplify: ⓐ \({(-5)}^{-2}\) ⓑ \(\text{-}{5}^{-2}\) ⓒ \({(-\frac{1}{5})}^{-2}\) ⓓ \(\text{-}{(\frac{1}{5})}^{-2}.\)

    Openbaar die antwoord

    ⓐ \(\frac{1}{25}\) ⓑ \(-\frac{1}{25}\) ⓒ 25 ⓓ \(-25\)

  15. Simplify: ⓐ \({(-7)}^{-2}\) ⓑ \(\text{-}{7}^{-2}\), ⓒ \({(-\frac{1}{7})}^{-2}\) ⓓ \(\text{-}{(\frac{1}{7})}^{-2}.\)

    Openbaar die antwoord

    ⓐ \(\frac{1}{49}\) ⓑ \(-\frac{1}{49}\) ⓒ 49 ⓓ \(-49\)

  16. Simplify: ⓐ \(4\cdot {2}^{-1}\) ⓑ \({(4\cdot 2)}^{-1}.\)

    Openbaar die antwoord
    ⓐ
    Do exponents before multiplication.    		\(4\cdot {2}^{-1}\)
    Use \({a}^{\text{-}n}=\frac{1}{{a}^{n}}\).\(4\cdot \frac{1}{{2}^{1}}\)
    Simplify.\(2\)
    ⓑ\({(4\cdot 2)}^{-1}\)
    Simplify inside the parentheses first.\({(8)}^{-1}\)
    Use \({a}^{\text{-}n}=\frac{1}{{a}^{n}}\).\(\frac{1}{{8}^{1}}\)
    Simplify.\(\frac{1}{8}\)

  17. Simplify: ⓐ \(6\cdot {3}^{-1}\) ⓑ \({(6\cdot 3)}^{-1}.\)

    Openbaar die antwoord

    ⓐ \(2\) ⓑ \(\frac{1}{18}\)

  18. Simplify: ⓐ \(8\cdot {2}^{-2}\) ⓑ \({(8\cdot 2)}^{-2}.\)

    Openbaar die antwoord

    ⓐ 2 ⓑ \(\frac{1}{256}\)

  19. Simplify: ⓐ \({x}^{-6}\) ⓑ \({({u}^{4})}^{-3}.\)

    Openbaar die antwoord
    1. ⓐ
      \(\begin{array}{llll} & & & \ {x}^{-6} \\ \text{Use the definition of a negative exponent,}\ {a}^{\text{-}n}=\frac{1}{{a}^{n}}. & & & \ \frac{1}{{x}^{6}}\end{array}\)
    2. ⓑ
      \(\begin{array}{llll} & & & \ {({u}^{4})}^{-3} \\ \text{Use the definition of a negative exponent,}\ {a}^{\text{-}n}=\frac{1}{{a}^{n}}. & & & \ \frac{1}{{({u}^{4})}^{3}} \\ \text{Simplify.} & & & \ \frac{1}{{u}^{12}}\end{array}\)

  20. Simplify: ⓐ \({y}^{-7}\) ⓑ \({({z}^{3})}^{-5}.\)

    Openbaar die antwoord

    ⓐ \(\frac{1}{{y}^{7}}\) ⓑ \(\frac{1}{{z}^{15}}\)

  21. Simplify: ⓐ \({p}^{-9}\) ⓑ \({({q}^{4})}^{-6}.\)

    Openbaar die antwoord

    ⓐ \(\frac{1}{{p}^{9}}\) ⓑ \(\frac{1}{{q}^{24}}\)

  22. Simplify: ⓐ \(5{y}^{-1}\) ⓑ \({(5y)}^{-1}\) ⓒ \({(-5y)}^{-1}.\)

    Openbaar die antwoord

    ⓐ

    \(5{y}^{-1}\)
    Notice the exponent applies to just the base y.
    Take the reciprocal of y and change the sign of the exponent.    		\(5\cdot \frac{1}{{y}^{1}}\)
    Simplify.\(\frac{5}{y}\)
    ⓑ
    \({(5y)}^{-1}\)
    Here the parentheses make the exponent apply to the base 5y.
    Take the reciprocal of 5y and change the sign of the exponent.    		\(\frac{1}{{(5y)}^{1}}\)
    Simplify.\(\frac{1}{5y}\)
    ⓒ
    \({(-5y)}^{-1}\)
    The base here is −5y.
    Take the reciprocal of −5y and change the sign of the exponent.    		\(\frac{1}{{(-5y)}^{1}}\)
    Simplify.\(\frac{1}{-5y}\)
    Use \(\frac{a}{\text{-}b}=-\frac{a}{b}.\)\(-\frac{1}{5y}\)

  23. Simplify: ⓐ \(8{p}^{-1}\) ⓑ \({(8p)}^{-1}\) ⓒ \({(-8p)}^{-1}.\)

    Openbaar die antwoord

    ⓐ \(\frac{8}{p}\) ⓑ \(\frac{1}{8p}\) ⓒ \(-\frac{1}{8p}\)

  24. Simplify: ⓐ \({11q}^{-1}\) ⓑ \({(11q)}^{-1}\)\(\text{-}{(11q)}^{-1}\) ⓒ \(-\frac{1}{11q}\)

    Openbaar die antwoord

    ⓐ \(\frac{1}{11q}\) ⓑ \(\frac{1}{11q}\)\(-\frac{1}{11q}\) ⓒ \(-\frac{11}{q}\)

  25. Simplify: ⓐ \({x}^{-4}\cdot {x}^{6}\) ⓑ \({y}^{-6}\cdot {y}^{4}\) ⓒ \({z}^{-5}\cdot {z}^{-3}.\)

    Openbaar die antwoord

    ⓐ

    \({x}^{-4}\cdot {x}^{6}\)
    Use the Product Property, \({a}^{m}\cdot {a}^{n}={a}^{m+n}.\)\({x}^{-4+6}\)
    Simplify.\({x}^{2}\)
    ⓑ
    \({y}^{-6}\cdot {y}^{4}\)
    Notice the same bases, so add the exponents.\({y}^{-6+4}\)
    Simplify.\({y}^{-2}\)
    Use the definition of a negative exponent, \({a}^{\text{-}n}=\frac{1}{{a}^{n}}.\)\(\frac{1}{{y}^{2}}\)
    ⓒ
    \({z}^{-5}\cdot {z}^{-3}\)
    Add the exponents, since the bases are the same.\({z}^{-5-3}\)
    Simplify.\({z}^{-8}\)
    Take the reciprocal and change the sign of the exponent,
    using the definition of a negative exponent.    		\(\frac{1}{{z}^{8}}\)

  26. Simplify: ⓐ \({x}^{-3}\cdot {x}^{7}\) ⓑ \({y}^{-7}\cdot {y}^{2}\) ⓒ \({z}^{-4}\cdot {z}^{-5}.\)

    Openbaar die antwoord

    ⓐ \({x}^{4}\) ⓑ \(\frac{1}{{y}^{5}}\) ⓒ \(\frac{1}{{z}^{9}}\)

  27. Simplify: ⓐ \({a}^{-1}\cdot {a}^{6}\) ⓑ \({b}^{-8}\cdot {b}^{4}\) ⓒ \({c}^{-8}\cdot {c}^{-7}.\)

    Openbaar die antwoord

    ⓐ \({a}^{5}\) ⓑ \(\frac{1}{{b}^{4}}\) ⓒ \(\frac{1}{{c}^{15}}\)

  28. Simplify: \(({m}^{4}{n}^{-3})({m}^{-5}{n}^{-2}).\)

    Openbaar die antwoord

    \(({m}^{4}{n}^{-3})({m}^{-5}{n}^{-2})\)
    Use the Commutative Property to get like bases together.\({m}^{4}{m}^{-5}\cdot {n}^{-2}{n}^{-3}\)
    Add the exponents for each base.\({m}^{-1}\cdot {n}^{-5}\)
    Take reciprocals and change the signs of the exponents.\(\frac{1}{{m}^{1}}\cdot \frac{1}{{n}^{5}}\)
    Simplify.\(\frac{1}{m{n}^{5}}\)

  29. Simplify: \(({p}^{6}{q}^{-2})({p}^{-9}{q}^{-1}).\)

    Openbaar die antwoord

    \(\frac{1}{{p}^{3}{q}^{3}}\)

  30. Simplify: \(({r}^{5}{s}^{-3})({r}^{-7}{s}^{-5}).\)

    Openbaar die antwoord

    \(\frac{1}{{r}^{2}{s}^{8}}\)

  31. Simplify: \((-6{c}^{-6}{d}^{4})(-5{c}^{-2}{d}^{-1}).\)

    Openbaar die antwoord

    \(\frac{30{d}^{3}}{{c}^{8}}\)

  32. Simplify: \({(6{k}^{3})}^{-2}.\)

    Openbaar die antwoord

    \({(6{k}^{3})}^{-2}\)
    Use the Product to a Power Property, \({(ab)}^{m}={a}^{m}{b}^{m}.\)\({(6)}^{-2}{({k}^{3})}^{-2}\)
    Use the Power Property, \({({a}^{m})}^{n}={a}^{m\cdot n}.\)\({6}^{-2}{k}^{-6}\)
    Use the Definition of a Negative Exponent, \({a}^{\text{-}n}=\frac{1}{{a}^{n}}.\)\(\frac{1}{{6}^{2}}\cdot \frac{1}{{k}^{6}}\)
    Simplify.\(\frac{1}{36{k}^{6}}\)

  33. Simplify: \({(-4{x}^{4})}^{-2}.\)

    Openbaar die antwoord

    \(\frac{1}{16{x}^{8}}\)

  34. Simplify: \({(2{b}^{3})}^{-4}.\)

    Openbaar die antwoord

    \(\frac{1}{16{b}^{12}}\)

  35. Simplify: \({(5{x}^{-3})}^{2}.\)

    Openbaar die antwoord

    \({(5{x}^{-3})}^{2}\)
    Use the Product to a Power Property, \({(ab)}^{m}={a}^{m}{b}^{m}.\)\({5}^{2}{({x}^{-3})}^{2}\)
    Simplify 52 and multiply the exponents of x using the Power
    Property, \({({a}^{m})}^{n}={a}^{m\cdot n}.\)    		\(25\cdot {x}^{-6}\)
    Rewrite x−6 by using the Definition of a Negative Exponent, \({a}^{\text{-}n}=\frac{1}{{a}^{n}}.\)\(25\cdot \frac{1}{{x}^{6}}\)
    Simplify.\(\frac{25}{{x}^{6}}\)

  36. Simplify: \({(8{a}^{-4})}^{2}.\)

    Openbaar die antwoord

    \(\frac{64}{{a}^{8}}\)

  37. Simplify: \({(2{c}^{-4})}^{3}.\)

    Openbaar die antwoord

    \(\frac{8}{{c}^{12}}\)

  38. Simplify: \(\frac{{r}^{5}}{{r}^{-4}}.\)

    Openbaar die antwoord

    \(\frac{{r}^{5}}{{r}^{-4}}\)
    Use the Quotient Property, \(\frac{{a}^{m}}{{a}^{n}}={a}^{m-n}.\)\({r}^{5-(-4)}\)
    Simplify.\({r}^{9}\)

  39. Simplify: \(\frac{{x}^{8}}{{x}^{-3}}.\)

    Openbaar die antwoord

    \({x}^{11}\)

  40. Simplify: \(\frac{{y}^{8}}{{y}^{-6}}.\)

    Openbaar die antwoord

    \({y}^{14}\)

Hou jou eie werk

'n Vry rekening voeg notas by oor elke les, 'n verslag van wat jy klaar gemaak het, jou opgelos probleme op een plek, en' n tugmeester wat jy kan vra oor hierdie bladsy. Die wiskunde self is oop vir almal, geteken in of nie.

Meld op Meld aan

Simbools wat hier gebruik word

Maak enige simbool van die volledige definisie,'n prent en wat elke letter daarin beteken.

Hoe om: Integer Exponents and Scientific Notation

  1. Use the definition of a negative exponent
  2. Simplify expressions with integer exponents
  3. Convert from decimal notation to scientific notation
  4. Convert scientific notation to decimal form
  5. Multiply and divide using scientific notation
  6. The power of
  7. The power of
  8. greater than 1, the power of 10 will be

Vrae wat mense vra

What does it mean to solve an equation?

To find every value of the unknown that makes both sides equal. Each step is an operation applied to both sides that keeps the solution set the same, until the unknown stands alone.

Why do I sometimes get two answers?

A quadratic can cross the axis twice, so it can have two solutions. A degree-n polynomial has up to n. The graph shows where each one comes from.

How do I know whether to factor or use the quadratic formula?

Try factoring for a few seconds: look for two numbers that multiply to a·c and add to b. If nothing obvious appears, the discriminant b² − 4ac tells you how many real roots there are, and the formula finds them without guessing.

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Linear equationsQuadratic equationsSystems of equationsInequalitiesFactoringExpandingSimplifying expressionsFunctions and graphsExponential and logarithmic equationsPolynomial equationsAbsolute value