Scientific Notation With Negative Exponent

Module 48: Integer Exponents and Scientific Notation

By the finish of this section, you will be able to:

Use the definition of a negative exponent
Simplify expressions with integer exponents
Catechumen from decimal annotation to scientific notation
Convert scientific notation to decimal form
Multiply and divide using scientific annotation

Use the Definition of a Negative Exponent

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

If $a$ is a real number, $a\ne 0$ , and $m\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}n$ are whole numbers, so

What if we but decrease 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.

$\begin{array}{c}\hfill \frac{{x}^{2}}{{x}^{5}}\hfill \\ \hfill {x}^{2-5}\hfill \\ \hfill {x}^{-3}\hfill \end{array}$

Nosotros can also simplify $\frac{{x}^{2}}{{x}^{5}}$ by dividing out common factors:

Illustrated in this figure is x times x divided by x times x times x times x times x. Two xes cancel out in the numerator and denominator. Below this is the simplified term: 1 divided by x cubed.

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

The negative exponent tells u.s. we tin can re-write the expression by taking the reciprocal of the base of operations and so irresolute the sign of the exponent.

Any expression that has negative exponents is non considered to be in simplest form. We will apply the definition of a negative exponent and other backdrop of exponents to write the expression with only positive exponents.

For example, if later on simplifying an expression we finish up with the expression ${x}^{-3}$ , we will take one more stride and write $\frac{1}{{x}^{3}}$ . The answer is considered to be in simplest class when it has simply positive exponents.

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

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

Testify answer

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

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

Show respond

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

In (Example 1) we raised an integer to a negative exponent. What happens when we raise a fraction to a negative exponent? We'll start by looking at what happens to a fraction whose numerator is one and whose denominator is an integer raised to a negative exponent.

	$\frac{1}{{a}^{\-n}}$
Use the definition of a negative exponent, ${a}^{-n}=\frac{1}{{a}^{n}}$ .	$\frac{1}{\frac{1}{{a}^{n}}}$
Simplify the complex fraction.	$1\cdot\frac{{a}^{n}}{1}$
Multiply.	${a}^{n}$

This leads to the Property of Negative Exponents.

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

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

Show answer

a) ${p}^{8}$ b) $64$

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

Show reply

a) ${q}^{7}$ b) $16$

Suppose at present we have a fraction raised to a negative exponent. Let'due south apply our definition of negative exponents to lead us to a new property.

	${\left(\frac{3}{4}\right)}^{-2}$
Use the definition of a negative exponent, ${a}^{-n}=\frac{1}{{a}^{n}}$ .	$\frac{1}{{\left(\frac{3}{4}\right)}^{2}}$
Simplify the denominator.	$\frac{1}{\frac{9}{16}}$
Simplify the complex fraction.	$\frac{16}{9}$
Just we know that $\frac{16}{9}$ is ${\left(\frac{4}{3}\right)}^{2}$ .
This tells us that:	${\left(\frac{3}{4}\right)}^{-2}={\left(\frac{4}{3}\right)}^{2}$

To get from the original fraction raised to a negative exponent to the final effect, we took the reciprocal of the base—the fraction—and changed the sign of the exponent.

This leads us to the Quotient to a Negative Power Holding.

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

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

Prove answer

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

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

Bear witness respond

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

When simplifying an expression with exponents, we must be careful to correctly place the base.

We must be conscientious to follow the Order of Operations. In the side by side example, parts (a) and (b) wait like, but the results are different.

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

Simplify: a) $6\cdot{3}^{-1}$ b) ${\left(6\cdot3\right)}^{-1}$ .

Show respond

a) $2$ b) $\frac{1}{18}$

Simplify: a) $8\cdot{2}^{-2}$ b) ${\left(8\cdot 2\right)}^{-2}$ .

Prove answer

a) 2 b) $\frac{1}{16}$

When a variable is raised to a negative exponent, we use the definition the same fashion we did with numbers. Nosotros will presume all variables are non-goose egg.

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

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

Show answer

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

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

Evidence answer

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

When at that place is a product and an exponent nosotros accept to be careful to use the exponent to the correct quantity. According to the Order of Operations, nosotros simplify expressions in parentheses before applying exponents. Nosotros'll see how this works in the next example.

With negative exponents, the Caliber Dominion needs only ane course $\frac{{a}^{m}}{{a}^{n}}={a}^{m-n}$ , for $a\ne 0$ . When the exponent in the denominator is larger than the exponent in the numerator, the exponent of the quotient will be negative.

Simplify Expressions with Integer Exponents

All of the exponent properties we developed earlier in the chapter with whole number exponents use to integer exponents, as well. We recapitulate them here for reference.

If $a\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}b$ are real numbers, and $m\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}n$ are integers, and so

$\begin{array}{cccccc}\mathbf{\text{Product Property}}\hfill & & & \hfill {a}^{m}\cdot{a}^{n}& =\hfill & {a}^{m+n}\hfill \\ \mathbf{\text{Power Property}}\hfill & & & \hfill {\left({a}^{m}\right)}^{n}& =\hfill & {a}^{m\cdot n}\hfill \\ \mathbf{\text{Product to a Power}}\hfill & & & \hfill {\left(ab\right)}^{m}& =\hfill & {a}^{m}{b}^{m}\hfill \\ \mathbf{\text{Quotient Property}}\hfill & & & \hfill \frac{{a}^{m}}{{a}^{n}}& =\hfill & {a}^{m-n},a\ne 0\hfill \\ \mathbf{\text{Zero Exponent Property}}\hfill & & & \hfill {a}^{0}& =\hfill & 1,a\ne 0\hfill \\ \mathbf{\text{Quotient to a Power Property}}\hfill & & & \hfill {\left(\frac{a}{b}\right)}^{m}& =\hfill & \frac{{a}^{m}}{{b}^{m}},\phantom{\rule{0.2em}{0ex}}\text{}\phantom{\rule{0.2em}{0ex}}b\ne 0\hfill \\ \mathbf{\text{Properties of Negative Exponents}}\hfill & & & \hfill {a}^{-n}& =\hfill & \frac{1}{{a}^{n}}\phantom{\rule{0.5em}{0ex}}\text{and}\phantom{\rule{0.5em}{0ex}}\frac{1}{{a}^{-n}}={a}^{n}\hfill \\ \mathbf{\text{Quotient to a Negative Exponent}}\hfill & & & \hfill {\left(\frac{a}{b}\right)}^{-n}& =\hfill & {\left(\frac{b}{a}\right)}^{n}\hfill \end{array}$

In the next two examples, we'll start by using the Commutative Property to grouping the aforementioned variables together. This makes it easier to identify the similar bases before using the Product Holding.

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

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

Testify reply

$\frac{1}{{p}^{3}{q}^{3}}$

Simplify: $\left({r}^{5}{s}^{-3}\right)\left({r}^{-7}{s}^{-5}\right)$ .

Show reply

$\frac{1}{{r}^{2}{s}^{8}}$

If the monomials have numerical coefficients, we multiply the coefficients, merely like nosotros did earlier.

Simplify: $\left(2{x}^{-6}{y}^{8}\right)\left(-5{x}^{5}{y}^{-3}\right)$ .

Simplify: $\left(3{u}^{-5}{v}^{7}\right)\left(-4{u}^{4}{v}^{-2}\right)$ .

Show answer

$-\frac{12{v}^{5}}{u}$

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

Bear witness respond

$\frac{30{d}^{3}}{{c}^{8}}$

In the next two examples, we'll use the Power Holding and the Product to a Power Belongings.

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

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

Show answer

$\frac{1}{16{x}^{8}}$

Simplify: ${\left(2{b}^{3}\right)}^{-4}$ .

Prove answer

$\frac{1}{16{b}^{12}}$

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

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

Evidence answer

$\frac{64}{{a}^{8}}$

Simplify: ${\left(2{c}^{-4}\right)}^{3}$ .

Prove answer

$\frac{8}{{c}^{12}}$

To simplify a fraction, nosotros utilize the Quotient Property and subtract the exponents.

Simplify: $\frac{{r}^{5}}{{r}^{-4}}$ .

Simplify: $\frac{{x}^{8}}{{x}^{-3}}$ .

Evidence answer

${x}^{11}$

Simplify: $\frac{{y}^{8}}{{y}^{-6}}$ .

Show answer

${y}^{13}$

Convert from Decimal Annotation to Scientific Note

Remember working with place value for whole numbers and decimals? Our number organization is based on powers of ten. Nosotros use tens, hundreds, thousands, and then on. Our decimal numbers are too 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 $4\phantom{\rule{0.2em}{0ex}}\times\phantom{\rule{0.2em}{0ex}}1,000$ and 0.004 means $4\phantom{\rule{0.2em}{0ex}}\times\phantom{\rule{0.2em}{0ex}}\frac{1}{1,000}$ .

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

$\begin{array}{cccc}4,000\hfill & & & \phantom{\rule{4em}{0ex}}0.004\hfill \\ 4\phantom{\rule{0.2em}{0ex}}\times\phantom{\rule{0.2em}{0ex}}1,000\hfill & & & \phantom{\rule{4em}{0ex}}4\phantom{\rule{0.2em}{0ex}}\times\phantom{\rule{0.2em}{0ex}}\frac{1}{1,000}\hfill \\ 4\phantom{\rule{0.2em}{0ex}}\times\phantom{\rule{0.2em}{0ex}}{10}^{3}\hfill & & & \phantom{\rule{4em}{0ex}}4\phantom{\rule{0.2em}{0ex}}\times\phantom{\rule{0.2em}{0ex}}\frac{1}{{10}^{3}}\hfill \\ & & & \phantom{\rule{4em}{0ex}}4\phantom{\rule{0.2em}{0ex}}\times\phantom{\rule{0.2em}{0ex}}{10}^{-3}\hfill \end{array}$

When a number is written every bit a product of two numbers, where the first cistron is a number greater than or equal to one but less than x, and the 2d factor is a power of 10 written in exponential form, it is said to be in scientific notation.

A number is expressed in scientific notation when it is of the form

$\begin{assortment}{cccc}& & & a\phantom{\rule{0.2em}{0ex}}\times\phantom{\rule{0.2em}{0ex}}{10}^{n}\phantom{\rule{0.2em}{0ex}}\text{where}\phantom{\rule{0.2em}{0ex}}one\le a<10\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}n\phantom{\rule{0.2em}{0ex}}\text{is an integer}\hfill \end{array}$

It is customary in scientific annotation to use as the $\phantom{\rule{0.2em}{0ex}}\times\phantom{\rule{0.2em}{0ex}}$ multiplication sign, fifty-fifty though nosotros avoid using this sign elsewhere in algebra.

If we look at what happened to the decimal betoken, we can come across a method to easily convert from decimal notation to scientific notation.

In both cases, the decimal was moved 3 places to become the first factor betwixt ane and 10

$\begin{array}{cccc}\text{The power of 10 is positive when the number is larger than 1:}\hfill & & & 4,000=4\phantom{\rule{0.2em}{0ex}}\times\phantom{\rule{0.2em}{0ex}}{10}^{3}\hfill \\ \text{The power of 10 is negative when the number is between 0 and 1:}\hfill & & & 0.004=4\phantom{\rule{0.2em}{0ex}}\times\phantom{\rule{0.2em}{0ex}}{10}^{-3}\hfill \end{array}$

How to Catechumen from Decimal Notation to Scientific Note

Write in scientific notation: 37,000.

Write in scientific annotation: $96,000$ .

Prove answer

$9.6\phantom{\rule{0.2em}{0ex}}\times\phantom{\rule{0.2em}{0ex}}{10}^{4}$

Write in scientific annotation: $48,300$ .

Show answer

$4.83\phantom{\rule{0.2em}{0ex}}\times\phantom{\rule{0.2em}{0ex}}{10}^{4}$

Move the decimal indicate so that the first factor is greater than or equal to one but less than 10.
Count the number of decimal places, northward, that the decimal indicate was moved.
Write the number as a product with a ability of 10.
If the original number is:
- greater than 1, the ability of 10 will exist 10^north.
- between 0 and 1, the ability of 10 will be 10⁻ⁿ.
Check.

Write in scientific note: $0.0052$ .

Write in scientific notation: $0.0078$ .

Prove reply

$7.8\phantom{\rule{0.2em}{0ex}}\times\phantom{\rule{0.2em}{0ex}}{10}^{-3}$

Write in scientific notation: $0.0129$ .

Evidence respond

$1.29\phantom{\rule{0.2em}{0ex}}\times\phantom{\rule{0.2em}{0ex}}{10}^{-2}$

Catechumen Scientific Note to Decimal Form

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

$\begin{array}{cccc}\hfill 9.12\phantom{\rule{0.2em}{0ex}}\times\phantom{\rule{0.2em}{0ex}}{10}^{4}\hfill & & & \hfill \phantom{\rule{4em}{0ex}}9.12\phantom{\rule{0.2em}{0ex}}\times\phantom{\rule{0.2em}{0ex}}{10}^{-4}\hfill \\ \hfill 9.12\phantom{\rule{0.2em}{0ex}}\times\phantom{\rule{0.2em}{0ex}}10,000\hfill & & & \hfill \phantom{\rule{4em}{0ex}}9.12\phantom{\rule{0.2em}{0ex}}\times\phantom{\rule{0.2em}{0ex}}0.0001\hfill \\ \hfill 91,200\hfill & & & \hfill \phantom{\rule{4em}{0ex}}0.000912\hfill \end{array}$

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.

$\begin{array}{cccc}\hfill 9.12\phantom{\rule{0.2em}{0ex}}\times\phantom{\rule{0.2em}{0ex}}{10}^{4}=91,200\hfill & & & \hfill \phantom{\rule{4em}{0ex}}9.12\phantom{\rule{0.2em}{0ex}}\times\phantom{\rule{0.2em}{0ex}}{10}^{-4}=0.000912\hfill \end{array}$

This figure has two columns. In the left column is 9.12 times 10 to the fourth power equals 91,200. Below this, the same scientific notation is repeated, with an arrow showing the decimal point in 9.12 being moved four places to the right. Because there are no digits after 2, the final two places are represented by blank spaces. Below this is the text

In both cases the decimal point moved iv 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

Convert to decimal form: $6.2\phantom{\rule{0.2em}{0ex}}\times\phantom{\rule{0.2em}{0ex}}{10}^{3}$ .

Convert to decimal class: $1.3\phantom{\rule{0.2em}{0ex}}\times\phantom{\rule{0.2em}{0ex}}{10}^{3}$ .

Catechumen to decimal grade: $9.25\phantom{\rule{0.2em}{0ex}}\times\phantom{\rule{0.2em}{0ex}}{10}^{4}$ .

The steps are summarized below.

To catechumen scientific notation to decimal form:

Decide the exponent, $n$ , on the factor ten.
Move the decimal $n$ places, calculation zeros if needed.
Check.

Catechumen to decimal grade: $8.9\phantom{\rule{0.2em}{0ex}}\times\phantom{\rule{0.2em}{0ex}}{10}^{-2}$ .

Convert to decimal form: $1.2\phantom{\rule{0.2em}{0ex}}\times\phantom{\rule{0.2em}{0ex}}{10}^{-4}$ .

Convert to decimal form: $7.5\phantom{\rule{0.2em}{0ex}}\times\phantom{\rule{0.2em}{0ex}}{10}^{-2}$ .

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 modest numbers to draw the size of an atom or the charge on an electron. When scientists perform calculations with very large or very pocket-size numbers, they use scientific notation. Scientific notation provides a manner 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 note.

Multiply. Write answers in decimal grade: $\left(4\phantom{\rule{0.2em}{0ex}}\times\phantom{\rule{0.2em}{0ex}}{10}^{5}\right)\left(2\phantom{\rule{0.2em}{0ex}}\times\phantom{\rule{0.2em}{0ex}}{10}^{-7}\right)$ .

Multiply $\left(3\phantom{\rule{0.2em}{0ex}}\times\phantom{\rule{0.2em}{0ex}}{10}^{6}\right)\left(2\phantom{\rule{0.2em}{0ex}}\times\phantom{\rule{0.2em}{0ex}}{10}^{-8}\right)$ . Write answers in decimal class.

Multiply $\left(3\phantom{\rule{0.2em}{0ex}}\times\phantom{\rule{0.2em}{0ex}}{10}^{-2}\right)\left(3\phantom{\rule{0.2em}{0ex}}\times\phantom{\rule{0.2em}{0ex}}{10}^{-1}\right)$ . Write answers in decimal form.

Split up. Write answers in decimal form: $\frac{9\phantom{\rule{0.2em}{0ex}}\times\phantom{\rule{0.2em}{0ex}}{10}^{3}}{3\phantom{\rule{0.2em}{0ex}}\times\phantom{\rule{0.2em}{0ex}}{10}^{-2}}$ .

Divide $\frac{8\phantom{\rule{0.2em}{0ex}}\times\phantom{\rule{0.2em}{0ex}}{10}^{4}}{2\phantom{\rule{0.2em}{0ex}}\times\phantom{\rule{0.2em}{0ex}}{10}^{-1}}$ . Write answers in decimal class.

Separate $\frac{8\phantom{\rule{0.2em}{0ex}}\times\phantom{\rule{0.2em}{0ex}}{10}^{2}}{4\phantom{\rule{0.2em}{0ex}}\times\phantom{\rule{0.2em}{0ex}}{10}^{-2}}$ . Write answers in decimal course.

Access these online resources for boosted instruction and practice with integer exponents and scientific notation:

Key Concepts

Practise exercises

Use the Definition of a Negative Exponent

In the post-obit exercises, simplify.

one. a) ${3}^{-4}$ b) ${10}^{-2}$	2. a) ${4}^{-2}$ b) ${10}^{-3}$
3. a) ${2}^{-8}$ b) ${10}^{-2}$	4. a) ${5}^{-3}$ b) ${10}^{-5}$
5. a) $\frac{1}{{c}^{-5}}$ b) $\frac{1}{{5}^{-2}}$	6. a) $\frac{1}{{c}^{-5}}$ b) $\frac{1}{{3}^{-2}}$
7. a) $\frac{1}{{t}^{-9}}$ b) $\frac{1}{{10}^{-4}}$	8. a) $\frac{1}{{q}^{-10}}$ b) $\frac{1}{{10}^{-3}}$
9. a) ${\left(\frac{3}{10}\right)}^{-2}$ b) ${\left(-\frac{2}{cd}\right)}^{-3}$	10. a) ${\left(\frac{5}{8}\right)}^{-2}$ b) ${\left(-\frac{3m}{n}\right)}^{-2}$
eleven. a) ${\left(\frac{7}{2}\right)}^{-3}$ b) ${\left(-\frac{3}{x{y}^{2}}\right)}^{-3}$	12. a) ${\left(\frac{4}{9}\right)}^{-3}$ b) ${\left(-\frac{{u}^{2}}{2v}\right)}^{-5}$
13. a) ${\left(-7\right)}^{-2}$ b) $-{7}^{-2}$ c) ${\left(-\frac{1}{7}\right)}^{-2}$ d) $-{\left(\frac{1}{7}\right)}^{-2}$	14. a) ${\left(-5\right)}^{-2}$ b) $-{5}^{-2}$ c) ${\left(-\frac{1}{5}\right)}^{-2}$ d) $-{\left(\frac{1}{5}\right)}^{-2}$
15. a) $-{5}^{-3}$ b) ${\left(-\frac{1}{5}\right)}^{-3}$ c) $-{\left(\frac{1}{5}\right)}^{-3}$ d) ${\left(-5\right)}^{-3}$	16. a) $-{3}^{-3}$ b) ${\left(-\frac{1}{3}\right)}^{-3}$ c) ${\left(\frac{1}{3}\right)}^{-3}$ d) ${\left(-3\right)}^{-3}$
17. a) $2\cdot{5}^{-1}$ b) ${\left(2\cdot 5\right)}^{-1}$	18. a) $3\cdot{5}^{-1}$ b) ${\left(3\cdot 5\right)}^{-1}$
nineteen. a) $3\cdot{4}^{-2}$ b) ${\left(3\cdot 4\right)}^{-2}$	20. a) $4\cdot{5}^{-2}$ b) ${\left(4\cdot 5\right)}^{-2}$
21. a) ${b}^{-5}$ b) ${\left({k}^{2}\right)}^{-5}$	22. a) ${m}^{-4}$ b) ${\left({x}^{3}\right)}^{-4}$
23. a) ${s}^{-8}$ b) ${\left({a}^{9}\right)}^{-10}$	24. a) ${p}^{-10}$ b) ${\left({q}^{6}\right)}^{-8}$
25. a) $6{r}^{-1}$ b) ${\left(6r\right)}^{-1}$ c) ${\left(-6r\right)}^{-1}$	26. a) $7{n}^{-1}$ b) ${\left(7n\right)}^{-1}$ c) ${\left(-7n\right)}^{-1}$
27. a) ${\left(2q\right)}^{-4}$ b) $2{q}^{-4}$ c) $-2{q}^{-4}$	28. a) ${\left(3p\right)}^{-2}$ b) $3{p}^{-2}$ c) $-3{p}^{-2}$

Convert from Decimal Note to Scientific Note

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

50. 57,000	51. 340,000
52. 8,750,000	53. 1,290,000
54. 0.026	55. 0.041
56. 0.00000871	57. 0.00000103

Convert Scientific Annotation to Decimal Grade

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

Multiply and Divide Using Scientific Notation

In the post-obit exercises, multiply. Write your answer in decimal course.

In the post-obit exercises, divide. Write your answer in decimal form.

Everyday Math

74. The population of the United States on July 1, 2010 was nearly 34,000,000. Write the number in scientific notation.	75. The population of the earth on July 1, 2010 was more than 6,850,000,000. Write the number in scientific notation
76. The average width of a human hair is 0.0018 centimetres. Write the number in scientific notation.	77. The probability of winning the 2010 Megamillions lottery was about 0.0000000057. Write the number in scientific annotation.
78. In 2010, the number of Facebook users each day who changed their status to 'engaged' was $2\phantom{\rule{0.2em}{0ex}}\times\phantom{\rule{0.2em}{0ex}}{10}^{4}$ . Convert this number to decimal form.	79. At the start of 2012, the United states of america federal upkeep had a deficit of more $\text{?}1.5\phantom{\rule{0.2em}{0ex}}\times\phantom{\rule{0.2em}{0ex}}{10}^{13}$ . Convert this number to decimal class.
80. The concentration of carbon dioxide in the atmosphere is $3.9\phantom{\rule{0.2em}{0ex}}\times\phantom{\rule{0.2em}{0ex}}{10}^{-4}$ . Convert this number to decimal form.	81. The width of a proton is $1\phantom{\rule{0.2em}{0ex}}\times\phantom{\rule{0.2em}{0ex}}{10}^{-5}$ of the width of an atom. Convert this number to decimal grade.
82. Health care costs The Centers for Medicare and Medicaid projects that American consumers volition spend more than $iv trillion on health care past 2017 Write iv trillion in decimal note. Write 4 trillion in scientific annotation.	83. Coin production In 1942, the U.S. Mint produced 154,500,000 nickels. Write 154,500,000 in scientific annotation.
84. Distance The altitude betwixt Earth and i of the brightest stars in the nighttime star is 33.7 light years. One light year is about six,000,000,000,000 (half dozen trillion), miles. a) Write the number of miles in 1 low-cal year in scientific notation. b)Use scientific notation to detect the altitude between Earth and the star in miles. Write the answer in scientific notation.	85. Debt At the end of fiscal yr 2019 the gross Canadian federal government debt was estimated to exist approximately $685,450,000,000 ($685.45 billion), according to the Federal Budget. The population of Canada was approximately 37,590,000 people at the end of financial twelvemonth 2019 a) Write the debt in scientific notation. b) Write the population in scientific annotation. c) Observe the amount of debt per person by using scientific notation to divide the debt past the population. Write the respond in scientific notation.

Writing Exercises.

86.

a) Explain the meaning of the exponent in the expression ${2}^{3}$ .

b) Explain the significant of the exponent in the expression ${2}^{-3}$ .

87. When you convert a number from decimal notation to scientific notation, how practice y'all know if the exponent will be positive or negative?

Answers

1. a) $\frac{1}{81}$ b) $\frac{1}{100}$	3. a) $\frac{1}{256}$ b) $\frac{1}{100}$	5. a) ${c}^{5}$ b) 25
vii. a) ${t}^{9}$ b) 10000	9. a) $\frac{100}{9}$ b) $-\frac{{c}^{3}{d}^{3}}{8}$	11. a) $\frac{8}{343}$ b) $-\frac{{x}^{3}{y}^{6}}{27}$
13. a) $\frac{1}{49}$ b) $-\frac{1}{49}$ c) 49 d) $-49$	15. a) $-\frac{1}{125}$ b) $-125$ c) $-125$ d) $-\frac{1}{125}$	17. a) $\frac{2}{5}$ b) $\frac{1}{10}$
19. a) $\frac{3}{16}$ b) $\frac{1}{144}$	21. a) $\frac{1}{{b}^{5}}$ b) $\frac{1}{{k}^{10}}$	23. a) $\frac{1}{{s}^{8}}$ b) $\frac{1}{{a}^{90}}$
25. a) $\frac{6}{r}$ b) $\frac{1}{6r}$ c) $-\frac{1}{6r}$	27. a) $\frac{1}{16{q}^{4}}$ b) $\frac{2}{{q}^{4}}$ c) $-\frac{2}{{q}^{4}}$	29. a) $\frac{1}{{s}^{4}}$ b) $\frac{1}{{q}^{5}}$ c) $\frac{1}{{y}^{7}}$
31. a) 1 b) ${y}^{6}$ c) $\frac{1}{{y}^{4}}$	33. $\frac{1}{x}$	35. $\frac{1}{{m}^{2}{n}^{4}}$
37. $\frac{1}{{p}^{5}{q}^{7}}$	39. $-\frac{14{k}^{5}}{{j}^{3}}$	41. $-\frac{40{n}^{3}}{m}$
43. $\frac{1}{64{y}^{9}}$	45. $\frac{4}{{p}^{10}}$	47. ${n}^{7}$
49. ${y}^{5}$	51. $3.4\phantom{\rule{0.2em}{0ex}}\times\phantom{\rule{0.2em}{0ex}}{10}^{5}$	53. $1.29\phantom{\rule{0.2em}{0ex}}\times\phantom{\rule{0.2em}{0ex}}{10}^{6}$
55. $4.1\phantom{\rule{0.2em}{0ex}}\times\phantom{\rule{0.2em}{0ex}}{10}^{-2}$	57. $1.03\phantom{\rule{0.2em}{0ex}}\times\phantom{\rule{0.2em}{0ex}}{10}^{-6}$	59. 830
61. 16,000,000,000	63. 0.038	65. 0.0000193
67. 0.02	69. $5.6\phantom{\rule{0.2em}{0ex}}\times\phantom{\rule{0.2em}{0ex}}{10}^{-6}$	71. 500,000,000
73. 20,000,000	75. $6.85\phantom{\rule{0.2em}{0ex}}\times\phantom{\rule{0.2em}{0ex}}{10}^{9}$ .	77. $5.7\phantom{\rule{0.2em}{0ex}}\times\phantom{\rule{0.2em}{0ex}}{10}^{-10}$
79. 15,000,000,000,000	81. 0.00001	83. $1.545\phantom{\rule{0.2em}{0ex}}\times\phantom{\rule{0.2em}{0ex}}{10}^{8}$
85. a) $1.86\phantom{\rule{0.2em}{0ex}}\times\phantom{\rule{0.2em}{0ex}}{10}^{13}$ b) $3\phantom{\rule{0.2em}{0ex}}\times\phantom{\rule{0.2em}{0ex}}{10}^{8}$ c) $6.2\phantom{\rule{0.2em}{0ex}}\times\phantom{\rule{0.2em}{0ex}}{10}^{4}$	87. Answers will vary

Attributions

This chapter has been adjusted from "Integer Exponents and Scientific Notation" in Uncomplicated Algebra (OpenStax) by Lynn Marecek and MaryAnne Anthony-Smith, which is under a CC By 4.0 Licence. Adapted past Izabela Mazur. See the Copyright page for more information.