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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:
- 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, \(n\), that the decimal point was moved.
- 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 \({10}^{\text{-}n}\)
- Check.
- To convert scientific notation to decimal form:
- Determine the exponent, \(n\), on the factor 10.
- 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.
- 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.
\(\text{12x}-4y-9x\)
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.
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:
- ⓐ \(\ {4}^{-2}\)
- ⓑ \(\ {10}^{-3}\)
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:
- ⓐ \(\ {(-3)}^{-2}\)
- ⓑ \(\ {-3}^{-2}\)
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:
- ⓐ \(\ {x}^{-4}\cdot {x}^{6}\)
- ⓑ \(\ {y}^{-6}\cdot {y}^{4}\)
- ⓒ \(\ {z}^{-5}\cdot {z}^{-3}\)
Solution
| ⓐ | |
| \({x}^{-4}\cdot {x}^{6}\) | |
| Use the Product Property, \({a}^{m}\cdot {a}^{n}={a}^{m+n}.\) | \({x}^{-4+6}\) |
| Simplify. | \({x}^{2}\) |
| ⓑ | |
| \({y}^{-6}\cdot {y}^{4}\) | |
| 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:
| |
| 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. | |
| 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:
- 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, \(n\), that the decimal point was moved.
- 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 \({10}^{-n}\).
- Check.
- Convert Scientific Notation to Decimal Form: To convert scientific notation to decimal form:
- Determine the exponent, \(n\), on the factor 10.
- 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.
- 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.
- ⓐ \(\ {(-6)}^{-2}\)
- ⓑ \(\ -{6}^{-2}\)
Try it.
- ⓐ \(\ {(-8)}^{-2}\)
- ⓑ \(\ -{8}^{-2}\)
Solution
- ⓐ \(\ \frac{1}{64}\)
- ⓑ \(\ -\frac{1}{64}\)
Try it.
- ⓐ \(\ {(-10)}^{-4}\)
- ⓑ \(\ -{10}^{-4}\)
Try it.
- ⓐ \(\ {(-4)}^{-6}\)
- ⓑ \(\ -{4}^{-6}\)
Solution
- ⓐ \(\ \frac{1}{4096}\)
- ⓑ \(\ -\frac{1}{4096}\)
Try it.
- ⓐ \(\ 5\cdot {2}^{-1}\)
- ⓑ \(\ {(5\cdot 2)}^{-1}\)
Try it.
- ⓐ \(\ 10\cdot {3}^{-1}\)
- ⓑ \(\ {(10\cdot 3)}^{-1}\)
Solution
- ⓐ \(\ \frac{10}{3}\)
- ⓑ \(\ \frac{1}{30}\)
Try it.
- ⓐ \(\ 4\cdot {10}^{-3}\)
- ⓑ \(\ {(4\cdot 10)}^{-3}\)
Try it.
- ⓐ \(\ 3\cdot {5}^{-2}\)
- ⓑ \(\ {(3\cdot 5)}^{-2}\)
Solution
- ⓐ \(\ \frac{3}{25}\)
- ⓑ \(\ \frac{1}{225}\)
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.
- ⓐ \(\ 4{x}^{-1}\)
- ⓑ \(\ {(4x)}^{-1}\)
- ⓒ \(\ {(-4x)}^{-1}\)
Try it.
- ⓐ \(\ 3{q}^{-1}\)
- ⓑ \(\ {(3q)}^{-1}\)
- ⓒ \(\ {(-3q)}^{-1}\)
Solution
- ⓐ \(\ \frac{3}{q}\)
- ⓑ \(\ \frac{1}{3q}\)
- ⓒ \(\ -\frac{1}{3q}\)
Try it.
- ⓐ \(\ 6{m}^{-1}\)
- ⓑ \(\ {(6m)}^{-1}\)
- ⓒ \(\ {(-6m)}^{-1}\)
Try it.
- ⓐ \(\ 10{k}^{-1}\)
- ⓑ \(\ {(10k)}^{-1}\)
- ⓒ \(\ {(-10k)}^{-1}\)
Solution
- ⓐ \(\ \frac{10}{k}\)
- ⓑ \(\ \frac{1}{10k}\)
- ⓒ \(\ -\frac{1}{10k}\)
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.
Now you Nijedan kalkulator ne slaže ovaj, ali komadiće su komputentni. Pokušajte jedan ispod, ili upišite vlastiti.
Vježba (40)
Pokušajte svaki na papiru prvo. Otkrijte odgovor provjeriti; provjereni mogu biti otvoreni u rješavač za svaki korak.
-
What is the place value of the \(6\) in the number \(64,891\)?
Otkrij odgovor
Ten thousands
-
Name the decimal: \(0.0012.\)
Otkrij odgovor
Twelve ten thousandths
-
Subtract: \(5-(-3).\)
Otkrij odgovor
\(8\)
-
Simplify: ⓐ \({4}^{-2}\) ⓑ \({10}^{-3}.\)
Otkrij odgovor
ⓐ \({4}^{-2}\) 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: ⓐ \({2}^{-3}\) ⓑ \({10}^{-7}.\)
Otkrij odgovor
ⓐ \(\frac{1}{8}\) ⓑ \(\frac{1}{{10}^{7}}\)
-
Simplify: ⓐ \({3}^{-2}\) ⓑ \({10}^{-4}.\)
Otkrij odgovor
ⓐ \(\frac{1}{9}\) ⓑ \(\frac{1}{10,000}\)
-
Simplify: ⓐ \(\frac{1}{{y}^{-4}}\) ⓑ \(\frac{1}{{3}^{-2}}.\)
Otkrij odgovor
ⓐ \(\frac{1}{{y}^{-4}}\) 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: ⓐ \(\frac{1}{{p}^{-8}}\) ⓑ \(\frac{1}{{4}^{-3}}.\)
Otkrij odgovor
ⓐ \({p}^{8}\) ⓑ \(64\)
-
Simplify: ⓐ \(\frac{1}{{q}^{-7}}\) ⓑ \(\frac{1}{{2}^{-4}}.\)
Otkrij odgovor
ⓐ \({q}^{7}\) ⓑ \(16\)
-
Simplify: ⓐ \({(\frac{5}{7})}^{-2}\) ⓑ \({(-\frac{2x}{y})}^{-3}.\)
Otkrij odgovor
ⓐ \({(\frac{5}{7})}^{-2}\) 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: ⓐ \({(\frac{2}{3})}^{-4}\) ⓑ \({(-\frac{6m}{n})}^{-2}.\)
Otkrij odgovor
ⓐ \(\frac{81}{16}\) ⓑ \(\frac{{n}^{2}}{36{m}^{2}}\)
-
Simplify: ⓐ \({(\frac{3}{5})}^{-3}\) ⓑ \({(-\frac{a}{2b})}^{-4}.\)
Otkrij odgovor
ⓐ \(\frac{125}{27}\) ⓑ \(\frac{16{b}^{4}}{{a}^{4}}\)
-
Simplify: ⓐ \({(-3)}^{-2}\) ⓑ \(\text{-}{3}^{-2}\) ⓒ \({(-\frac{1}{3})}^{-2}\) ⓓ \(\text{-}{(\frac{1}{3})}^{-2}.\)
Otkrij odgovor
ⓐ 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: ⓐ \({(-5)}^{-2}\) ⓑ \(\text{-}{5}^{-2}\) ⓒ \({(-\frac{1}{5})}^{-2}\) ⓓ \(\text{-}{(\frac{1}{5})}^{-2}.\)
Otkrij odgovor
ⓐ \(\frac{1}{25}\) ⓑ \(-\frac{1}{25}\) ⓒ 25 ⓓ \(-25\)
-
Simplify: ⓐ \({(-7)}^{-2}\) ⓑ \(\text{-}{7}^{-2}\), ⓒ \({(-\frac{1}{7})}^{-2}\) ⓓ \(\text{-}{(\frac{1}{7})}^{-2}.\)
Otkrij odgovor
ⓐ \(\frac{1}{49}\) ⓑ \(-\frac{1}{49}\) ⓒ 49 ⓓ \(-49\)
-
Simplify: ⓐ \(4\cdot {2}^{-1}\) ⓑ \({(4\cdot 2)}^{-1}.\)
Otkrij odgovor
ⓐ
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: ⓐ \(6\cdot {3}^{-1}\) ⓑ \({(6\cdot 3)}^{-1}.\)
Otkrij odgovor
ⓐ \(2\) ⓑ \(\frac{1}{18}\)
-
Simplify: ⓐ \(8\cdot {2}^{-2}\) ⓑ \({(8\cdot 2)}^{-2}.\)
Otkrij odgovor
ⓐ 2 ⓑ \(\frac{1}{256}\)
-
Simplify: ⓐ \({x}^{-6}\) ⓑ \({({u}^{4})}^{-3}.\)
Otkrij odgovor
- ⓐ
\(\begin{array}{llll} & & & \ {x}^{-6} \\ \text{Use the definition of a negative exponent,}\ {a}^{\text{-}n}=\frac{1}{{a}^{n}}. & & & \ \frac{1}{{x}^{6}}\end{array}\) - ⓑ
\(\begin{array}{llll} & & & \ {({u}^{4})}^{-3} \\ \text{Use the definition of a negative exponent,}\ {a}^{\text{-}n}=\frac{1}{{a}^{n}}. & & & \ \frac{1}{{({u}^{4})}^{3}} \\ \text{Simplify.} & & & \ \frac{1}{{u}^{12}}\end{array}\)
- ⓐ
-
Simplify: ⓐ \({y}^{-7}\) ⓑ \({({z}^{3})}^{-5}.\)
Otkrij odgovor
ⓐ \(\frac{1}{{y}^{7}}\) ⓑ \(\frac{1}{{z}^{15}}\)
-
Simplify: ⓐ \({p}^{-9}\) ⓑ \({({q}^{4})}^{-6}.\)
Otkrij odgovor
ⓐ \(\frac{1}{{p}^{9}}\) ⓑ \(\frac{1}{{q}^{24}}\)
-
Simplify: ⓐ \(5{y}^{-1}\) ⓑ \({(5y)}^{-1}\) ⓒ \({(-5y)}^{-1}.\)
Otkrij odgovor
ⓐ
ⓑ\(5{y}^{-1}\) 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: ⓐ \(8{p}^{-1}\) ⓑ \({(8p)}^{-1}\) ⓒ \({(-8p)}^{-1}.\)
Otkrij odgovor
ⓐ \(\frac{8}{p}\) ⓑ \(\frac{1}{8p}\) ⓒ \(-\frac{1}{8p}\)
-
Simplify: ⓐ \({11q}^{-1}\) ⓑ \({(11q)}^{-1}\)\(\text{-}{(11q)}^{-1}\) ⓒ \(-\frac{1}{11q}\)
Otkrij odgovor
ⓐ \(\frac{1}{11q}\) ⓑ \(\frac{1}{11q}\)\(-\frac{1}{11q}\) ⓒ \(-\frac{11}{q}\)
-
Simplify: ⓐ \({x}^{-4}\cdot {x}^{6}\) ⓑ \({y}^{-6}\cdot {y}^{4}\) ⓒ \({z}^{-5}\cdot {z}^{-3}.\)
Otkrij odgovor
ⓐ
ⓑ\({x}^{-4}\cdot {x}^{6}\) 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: ⓐ \({x}^{-3}\cdot {x}^{7}\) ⓑ \({y}^{-7}\cdot {y}^{2}\) ⓒ \({z}^{-4}\cdot {z}^{-5}.\)
Otkrij odgovor
ⓐ \({x}^{4}\) ⓑ \(\frac{1}{{y}^{5}}\) ⓒ \(\frac{1}{{z}^{9}}\)
-
Simplify: ⓐ \({a}^{-1}\cdot {a}^{6}\) ⓑ \({b}^{-8}\cdot {b}^{4}\) ⓒ \({c}^{-8}\cdot {c}^{-7}.\)
Otkrij odgovor
ⓐ \({a}^{5}\) ⓑ \(\frac{1}{{b}^{4}}\) ⓒ \(\frac{1}{{c}^{15}}\)
-
Simplify: \(({m}^{4}{n}^{-3})({m}^{-5}{n}^{-2}).\)
Otkrij odgovor
\(({m}^{4}{n}^{-3})({m}^{-5}{n}^{-2})\) 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: \(({p}^{6}{q}^{-2})({p}^{-9}{q}^{-1}).\)
Otkrij odgovor
\(\frac{1}{{p}^{3}{q}^{3}}\)
-
Simplify: \(({r}^{5}{s}^{-3})({r}^{-7}{s}^{-5}).\)
Otkrij odgovor
\(\frac{1}{{r}^{2}{s}^{8}}\)
-
Simplify: \((-6{c}^{-6}{d}^{4})(-5{c}^{-2}{d}^{-1}).\)
Otkrij odgovor
\(\frac{30{d}^{3}}{{c}^{8}}\)
-
Simplify: \({(6{k}^{3})}^{-2}.\)
Otkrij odgovor
\({(6{k}^{3})}^{-2}\) 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: \({(-4{x}^{4})}^{-2}.\)
Otkrij odgovor
\(\frac{1}{16{x}^{8}}\)
-
Simplify: \({(2{b}^{3})}^{-4}.\)
Otkrij odgovor
\(\frac{1}{16{b}^{12}}\)
-
Simplify: \({(5{x}^{-3})}^{2}.\)
Otkrij odgovor
\({(5{x}^{-3})}^{2}\) 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: \({(8{a}^{-4})}^{2}.\)
Otkrij odgovor
\(\frac{64}{{a}^{8}}\)
-
Simplify: \({(2{c}^{-4})}^{3}.\)
Otkrij odgovor
\(\frac{8}{{c}^{12}}\)
-
Simplify: \(\frac{{r}^{5}}{{r}^{-4}}.\)
Otkrij odgovor
\(\frac{{r}^{5}}{{r}^{-4}}\) Use the Quotient Property, \(\frac{{a}^{m}}{{a}^{n}}={a}^{m-n}.\) \({r}^{5-(-4)}\) Simplify. \({r}^{9}\) -
Simplify: \(\frac{{x}^{8}}{{x}^{-3}}.\)
Otkrij odgovor
\({x}^{11}\)
-
Simplify: \(\frac{{y}^{8}}{{y}^{-6}}.\)
Otkrij odgovor
\({y}^{14}\)
Besplatan račun dodaje bilješke na svakoj lekciji, zapis onoga što ste završili, rješeni problemi na jednom mjestu, i tutor možete pitati o ovoj stranici. Sama matematika je otvorena za sve, potpisani ili ne.
Prijavi se PrijavaSimboli koji se koriste ovdje
Dodirnite bilo koji simbol za punu definiciju, sliku, i što znači svako slovo u njoj.
Kako: 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
- The power of
- The power of
- greater than 1, the power of 10 will be
Pitanja koja ljudi postavljaju
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.
Dijelovi ove stranice prilagođeni su od OpenStax Elementary Algebra 2e (CC BY-NC-SA 4.0), OpenStax Prealgebra 2e (CC BY-NC-SA 4.0). Ovdje su kondenzirane i objašnjene; greške su naše. Budući da je njegov izvor licenciran za nekomercijalnu upotrebu, ova lekcija je besplatna, ne prodaje ništa, i dijeli se pod istoj licencom.
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