maths.free › Algebra › 6. Polynomials › Integer Exponents and Scientific Notation

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.

Summary of Exponent Properties

If \(a\ \text{and}\ b\) are real numbers, and \(m\ \text{and}\ n\) are integers, then

\[\begin{array}{llllll}\text{Product Property} & & & {a}^{m}\cdot {a}^{n} & = & {a}^{m+n} \\ \text{Power Property} & & & {({a}^{m})}^{n} & = & {a}^{m\cdot n} \\ \text{Product to a Power} & & & {(ab)}^{m} & = & {a}^{m}{b}^{m} \\ \text{Quotient Property} & & & \frac{{a}^{m}}{{a}^{n}} & = & {a}^{m-n},a\ne 0 \\ \text{Zero Exponent Property} & & & {a}^{0} & = & 1,a\ne 0 \\ \text{Quotient to a Power Property} & & & {(\frac{a}{b})}^{m} & = & \frac{{a}^{m}}{{b}^{m}},\ \ b\ne 0 \\ \text{Properties of Negative Exponents} & & & {a}^{\text{-}n} & = & \frac{1}{{a}^{n}}\ \text{and}\ \frac{1}{{a}^{\text{-}n}}={a}^{n} \\ \text{Quotient to a Negative Exponent} & & & {(\frac{a}{b})}^{\text{-}n} & = & {(\frac{b}{a})}^{n}\end{array}\]

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 see.

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

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.

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

Try it.

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

\(-9c-18c\)

Solution

\(-27c\)

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:

ⓐ \(y=5\)
ⓑ \(y=-1\)
ⓒ \(y=0\)

Try it.

Evaluate \(10-12x\) when:

ⓐ \(x=3\)
ⓑ \(x=0\)
ⓒ \(x=-1\)

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.

Practice (40)

Try each one on paper first. Reveal the answer to check; verified ones can be opened in the solver for every step.

What is the place value of the \(6\) in the number \(64,891\)?
If you missed this problem, review .

Giải đáp

Ten thousands
Name the decimal: \(0.0012.\)
If you missed this problem, review .

Giải đáp

Twelve ten thousandths
Subtract: \(5-(-3).\)
If you missed this problem, review .

Giải đáp

\(8\)

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

Giải đáp

ⓐ	\({4}^{-2}\)
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.	\(\frac{1}{1000}\)

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

Giải đáp

ⓐ \(\frac{1}{8}\) ⓑ \(\frac{1}{{10}^{7}}\)
Simplify: ⓐ \({3}^{-2}\) ⓑ \({10}^{-4}.\)

Giải đáp

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

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

Giải đáp

ⓐ	\(\frac{1}{{y}^{-4}}\)
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.	\(9\)

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

Giải đáp

ⓐ \({p}^{8}\) ⓑ \(64\)
Simplify: ⓐ \(\frac{1}{{q}^{-7}}\) ⓑ \(\frac{1}{{2}^{-4}}.\)

Giải đáp

ⓐ \({q}^{7}\) ⓑ \(16\)

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

Giải đáp

ⓐ	\({(\frac{5}{7})}^{-2}\)
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.	\(-\frac{{y}^{3}}{8{x}^{3}}\)

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

Giải đáp

ⓐ \(\frac{81}{16}\) ⓑ \(\frac{{n}^{2}}{36{m}^{2}}\)
Simplify: ⓐ \({(\frac{3}{5})}^{-3}\) ⓑ \({(-\frac{a}{2b})}^{-4}.\)

Giải đáp

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

Giải đáp

ⓐ 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.	\(-9\)

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

Giải đáp

ⓐ \(\frac{1}{25}\) ⓑ \(-\frac{1}{25}\) ⓒ 25 ⓓ \(-25\)
Simplify: ⓐ \({(-7)}^{-2}\) ⓑ \(\text{-}{7}^{-2}\), ⓒ \({(-\frac{1}{7})}^{-2}\) ⓓ \(\text{-}{(\frac{1}{7})}^{-2}.\)

Giải đáp

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

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

Giải đáp

ⓐ 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.	\(\frac{1}{8}\)

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

Giải đáp

ⓐ \(2\) ⓑ \(\frac{1}{18}\)
Simplify: ⓐ \(8\cdot {2}^{-2}\) ⓑ \({(8\cdot 2)}^{-2}.\)

Giải đáp

ⓐ 2 ⓑ \(\frac{1}{256}\)
Simplify: ⓐ \({x}^{-6}\) ⓑ \({({u}^{4})}^{-3}.\)
Giải đáp
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. ⓑ
  \(\begin{array}{llll} & & & \ {({u}^{4})}^{-3} \\ \text{Use the definition of a negative exponent,}\ {a}^{\text{-}n}=\frac{1}{{a}^{n}}. & & & \ \frac{1}{{({u}^{4})}^{3}} \\ \text{Simplify.} & & & \ \frac{1}{{u}^{12}}\end{array}\)
Simplify: ⓐ \({y}^{-7}\) ⓑ \({({z}^{3})}^{-5}.\)

Giải đáp

ⓐ \(\frac{1}{{y}^{7}}\) ⓑ \(\frac{1}{{z}^{15}}\)
Simplify: ⓐ \({p}^{-9}\) ⓑ \({({q}^{4})}^{-6}.\)

Giải đáp

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

Giải đáp

ⓐ

	\(5{y}^{-1}\)
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 \(\frac{a}{\text{-}b}=-\frac{a}{b}.\)	\(-\frac{1}{5y}\)

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

Giải đáp

ⓐ \(\frac{8}{p}\) ⓑ \(\frac{1}{8p}\) ⓒ \(-\frac{1}{8p}\)
Simplify: ⓐ \({11q}^{-1}\) ⓑ \({(11q)}^{-1}\)\(\text{-}{(11q)}^{-1}\) ⓒ \(-\frac{1}{11q}\)

Giải đáp

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

Giải đáp

ⓐ

	\({x}^{-4}\cdot {x}^{6}\)
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.	\(\frac{1}{{z}^{8}}\)

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

Giải đáp

ⓐ \({x}^{4}\) ⓑ \(\frac{1}{{y}^{5}}\) ⓒ \(\frac{1}{{z}^{9}}\)
Simplify: ⓐ \({a}^{-1}\cdot {a}^{6}\) ⓑ \({b}^{-8}\cdot {b}^{4}\) ⓒ \({c}^{-8}\cdot {c}^{-7}.\)

Giải đáp

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

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

Giải đáp

	\(({m}^{4}{n}^{-3})({m}^{-5}{n}^{-2})\)
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.	\(\frac{1}{m{n}^{5}}\)

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

Giải đáp

\(\frac{1}{{p}^{3}{q}^{3}}\)
Simplify: \(({r}^{5}{s}^{-3})({r}^{-7}{s}^{-5}).\)

Giải đáp

\(\frac{1}{{r}^{2}{s}^{8}}\)
Simplify: \((-6{c}^{-6}{d}^{4})(-5{c}^{-2}{d}^{-1}).\)

Giải đáp

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

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

Giải đáp

	\({(6{k}^{3})}^{-2}\)
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.	\(\frac{1}{36{k}^{6}}\)

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

Giải đáp

\(\frac{1}{16{x}^{8}}\)
Simplify: \({(2{b}^{3})}^{-4}.\)

Giải đáp

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

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

Giải đáp

	\({(5{x}^{-3})}^{2}\)
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.	\(\frac{25}{{x}^{6}}\)

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

Giải đáp

\(\frac{64}{{a}^{8}}\)
Simplify: \({(2{c}^{-4})}^{3}.\)

Giải đáp

\(\frac{8}{{c}^{12}}\)
Simplify: \(\frac{{r}^{5}}{{r}^{-4}}.\)

Giải đáp

\(\frac{{r}^{5}}{{r}^{-4}}\)
Use the Quotient Property, \(\frac{{a}^{m}}{{a}^{n}}={a}^{m-n}.\) \({r}^{5-(-4)}\)
Simplify. \({r}^{9}\)
Simplify: \(\frac{{x}^{8}}{{x}^{-3}}.\)

Giải đáp

\({x}^{11}\)
Simplify: \(\frac{{y}^{8}}{{y}^{-6}}.\)

Giải đáp

\({y}^{14}\)

Symbols used here

\leq,\ \geq

less/greater than or equal
Inequalities that allow equality; < and > exclude it.

\neq

not equal
The two sides are different.

\pm

plus or minus
Both signs at once: x = 3 ± 2 means 5 and 1.

\sqrt{x},\ \sqrt[n]{x}

square root, n-th root
The non-negative number whose square (n-th power) is x.

|x|

absolute value / modulus
Distance from zero: |−3| = 3. For a complex number, distance from the origin.

imaginary unit
i² = −1.

\log_b x,\ \ln x

logarithm, natural log
The exponent b must be raised to for x; ln uses base e.

\mathbb{N},\ \mathbb{Z},\ \mathbb{Q},\ \mathbb{R},\ \mathbb{C}

number sets
Naturals, integers, rationals, reals, complex numbers.

How to: Integer Exponents and Scientific Notation

Use the definition of a negative exponent
Simplify expressions with integer exponents
Convert from decimal notation to scientific notation
Convert scientific notation to decimal form
Multiply and divide using scientific notation
Move the decimal point so that the first factor is greater than or equal to 1 but less than 10.
Count the number of decimal places,
Write the number as a product with a power of 10.

Questions people ask

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.

Thử đi.

Parts of this page are adapted from OpenStax Elementary Algebra 2e (CC BY-NC-SA 4.0). Condensed and re-explained here; errors are ours.

More in Algebra

Linear equations Quadratic equations Systems of equations Inequalities Factoring Expanding Simplifying expressions Functions and graphs Exponential and logarithmic equations Polynomial equations Absolute value

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