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Integer Exponents and Scientific Notation

Use the definition of a negative exponent

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: 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 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.

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.

ⓐ $\ {(-6)}^{-2}$
ⓑ $\ -{6}^{-2}$

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.

45,000

Try it.

280,000

Solution

2.8 × 10⁵

Try it.

8,750,000

Try it.

1,290,000

Solution

1.29 × 10⁶

Try it.

0.036

Try it.

0.041

Solution

4.1 × 10⁻²

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.

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 .

ເປີດເຜີຍຄຳຕອບ

ten thousand
Name the decimal $0.0012.$
If you missed this problem, review .

ເປີດເຜີຍຄຳຕອບ

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

ເປີດເຜີຍຄຳຕອບ

$8$
Simplify:
1. ⓐ $\ {4}^{-2}$
2. ⓑ $\ {10}^{-3}$
ເປີດເຜີຍຄຳຕອບ

ⓐ
${4}^{-2}$
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. $\frac{1}{1000}$
Simplify:
1. ⓐ $\ {2}^{-3}$
2. ⓑ $\ {10}^{-2}$
ເປີດເຜີຍຄຳຕອບ
1. ⓐ $\ \frac{1}{8}$
2. ⓑ $\ \frac{1}{100}$
Simplify:
1. ⓐ $\ {3}^{-2}$
2. ⓑ $\ {10}^{-4}$
ເປີດເຜີຍຄຳຕອບ
1. ⓐ $\ \frac{1}{9}$
2. ⓑ $\ \frac{1}{10,000}$

Simplify:

ⓐ $\ {(-3)}^{-2}$
ⓑ $\ {-3}^{-2}$

ເປີດເຜີຍຄຳຕອບ

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.	$-\frac{1}{9}$

Simplify:
1. ⓐ $\ {(-5)}^{-2}$
2. ⓑ $\ -{5}^{-2}$
ເປີດເຜີຍຄຳຕອບ
1. ⓐ $\ \frac{1}{25}$
2. ⓑ $\ -\frac{1}{25}$
Simplify:
1. ⓐ $\ {(-2)}^{-2}$
2. ⓑ $\ {-2}^{-2}$
ເປີດເຜີຍຄຳຕອບ
1. ⓐ $\ \frac{1}{4}$
2. ⓑ $\ -\frac{1}{4}$
Simplify:
1. ⓐ $\ 4\cdot {2}^{-1}$
2. ⓑ $\ {(4\cdot 2)}^{-1}$
ເປີດເຜີຍຄຳຕອບ

Remember to always follow the order of operations.
ⓐ
Do exponents before multiplication. $4\cdot {2}^{-1}$
Use ${a}^{-n}=\frac{1}{{a}^{n}}.$ $4\cdot \frac{1}{{2}^{1}}$
Simplify. $2$
ⓑ ${(4\cdot 2)}^{-1}$
Simplify inside the parentheses first. ${(8)}^{-1}$
Use ${a}^{-n}=\frac{1}{{a}^{n}}.$ $\frac{1}{{8}^{1}}$
Simplify. $\frac{1}{8}$
Simplify:
1. ⓐ $\ 6\cdot {3}^{-1}$
2. ⓑ $\ {(6\cdot 3)}^{-1}$
ເປີດເຜີຍຄຳຕອບ
1. ⓐ $\ 2$
2. ⓑ $\ \frac{1}{18}$
Simplify:
1. ⓐ $\ 8\cdot {2}^{-2}$
2. ⓑ $\ {(8\cdot 2)}^{-2}$
ເປີດເຜີຍຄຳຕອບ
1. ⓐ $\ 2$
2. ⓑ $\ \frac{1}{256}$
Simplify: ${x}^{-6}.$

ເປີດເຜີຍຄຳຕອບ

${x}^{-6}$
Use the definition of a negative exponent, ${a}^{-n}=\frac{1}{{a}^{n}}.$ $\frac{1}{{x}^{6}}$
Simplify: ${y}^{-7}.$

ເປີດເຜີຍຄຳຕອບ

$\frac{1}{{y}^{7}}$
Simplify: ${z}^{-8}.$

ເປີດເຜີຍຄຳຕອບ

$\frac{1}{{z}^{8}}$

Simplify:

ⓐ $\ 5{y}^{-1}$
ⓑ $\ {(5y)}^{-1}$

ເປີດເຜີຍຄຳຕອບ

ⓐ
Notice the exponent applies to just the base $y$.	$5{y}^{-1}$
Take the reciprocal of $y$ and change the sign of the exponent.	$5\cdot \frac{1}{{y}^{1}}$
Simplify.	$\frac{5}{y}$

ⓑ
Here the parentheses make the exponent apply to the base $5y$.	${(5y)}^{-1}$
Take the reciprocal of $5y$ and change the sign of the exponent.	$\frac{1}{{(5y)}^{1}}$
Simplify.	$\frac{1}{5y}$

ⓒ
	${(-5y)}^{-1}$
The base is $-5y$. Take the reciprocal of $-5y$ and change the sign of the exponent.	$\frac{1}{{(-5y)}^{1}}$
Simplify.	$\frac{1}{-5y}$
Use $\frac{a}{-b}=-\frac{a}{b}.$	$-\frac{1}{5y}$

Simplify:
1. ⓐ $\ 8{p}^{-1}$
2. ⓑ $\ {(8p)}^{-1}$
3. ⓒ $\ {(-8p)}^{-1}$
ເປີດເຜີຍຄຳຕອບ
1. ⓐ $\ \frac{8}{p}$
2. ⓑ $\ \frac{1}{8p}$
3. ⓒ $\ -\frac{1}{8p}$
Simplify:
1. ⓐ $\ 11{q}^{-1}$
2. ⓑ $\ {(11q)}^{-1}$
3. ⓒ $\ {(-11q)}^{-1}$
ເປີດເຜີຍຄຳຕອບ
1. ⓐ $\ \frac{11}{q}$
2. ⓑ $\ \frac{1}{11q}$
3. ⓒ $\ -\frac{1}{11q}$

Simplify:

ⓐ $\ {x}^{-4}\cdot {x}^{6}$
ⓑ $\ {y}^{-6}\cdot {y}^{4}$

ເປີດເຜີຍຄຳຕອບ

ⓐ
	${x}^{-4}\cdot {x}^{6}$
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, ${a}^{-n}=\frac{1}{{a}^{n}}.$	$\frac{1}{{z}^{8}}$

Simplify:
1. ⓐ $\ {x}^{-3}\cdot {x}^{7}$
2. ⓑ $\ {y}^{-7}\cdot {y}^{2}$
3. ⓒ $\ {z}^{-4}\cdot {z}^{-5}$
ເປີດເຜີຍຄຳຕອບ
1. ⓐ $\ {x}^{4}$
2. ⓑ $\ \frac{1}{{y}^{5}}$
3. ⓒ $\ \frac{1}{{z}^{9}}$
Simplify:
1. ⓐ $\ {a}^{-1}\cdot {a}^{6}$
2. ⓑ $\ {b}^{-8}\cdot {b}^{4}$
3. ⓒ $\ {c}^{-8}\cdot {c}^{-7}$
ເປີດເຜີຍຄຳຕອບ
1. ⓐ $\ {a}^{5}$
2. ⓑ $\ \frac{1}{{b}^{4}}$
3. ⓒ $\ \frac{1}{{c}^{15}}$

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

ເປີດເຜີຍຄຳຕອບ

	$({m}^{4}{n}^{-3})({m}^{-5}{n}^{-2})$
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: $({p}^{6}{q}^{-2})({p}^{-9}{q}^{-1}).$

ເປີດເຜີຍຄຳຕອບ

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

ເປີດເຜີຍຄຳຕອບ

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

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

ເປີດເຜີຍຄຳຕອບ

	$(2{x}^{-6}{y}^{8})(-5{x}^{5}{y}^{-3})$
Rewrite with the like bases together.	$2(-5)\cdot ({x}^{-6}{x}^{5})\cdot ({y}^{8}{y}^{-3})$
Simplify.	$-10\cdot {x}^{-1}\cdot {y}^{5}$
Use the definition of a negative exponent, ${a}^{-n}=\frac{1}{{a}^{n}}.$	$-10\cdot \frac{1}{{x}^{1}}\cdot {y}^{5}$
Simplify.	$\frac{-10{y}^{5}}{x}$

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

ເປີດເຜີຍຄຳຕອບ

$-\frac{12{v}^{5}}{u}$
Simplify: $(-6{c}^{-6}{d}^{4})(-5{c}^{-2}{d}^{-1}).$

ເປີດເຜີຍຄຳຕອບ

$\frac{30{d}^{3}}{{c}^{8}}$
Simplify: ${({k}^{3})}^{-2}.$

ເປີດເຜີຍຄຳຕອບ

${({k}^{3})}^{-2}$
Use the Product to a Power Property, ${(ab)}^{m}={a}^{m}{b}^{m}.$ ${k}^{3(-2)}$
Simplify. ${k}^{-6}$
Rewrite with a positive exponent. $\frac{1}{{k}^{6}}$
Simplify: ${({x}^{4})}^{-1}.$

ເປີດເຜີຍຄຳຕອບ

$\frac{1}{{x}^{4}}$
Simplify: ${({y}^{2})}^{-2}.$

ເປີດເຜີຍຄຳຕອບ

$\frac{1}{{y}^{4}}$

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

ເປີດເຜີຍຄຳຕອບ

	${(5{x}^{-3})}^{2}$
Use the Product to a Power Property, ${(ab)}^{m}={a}^{m}{b}^{m}.$	${5}^{2}{({x}^{-3})}^{2}$
Simplify ${5}^{2}$ and multiply the exponents of $x$ using the Power Property, ${({a}^{m})}^{n}={a}^{m\cdot n}.$	$25{x}^{-6}$
Rewrite ${x}^{-6}$ by using the definition of a negative exponent, ${a}^{-n}=\frac{1}{{a}^{n}}.$	$25\cdot \frac{1}{{x}^{6}}$
Simplify	$\frac{25}{{x}^{6}}$

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

ເປີດເຜີຍຄຳຕອບ

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

ເປີດເຜີຍຄຳຕອບ

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

ເປີດເຜີຍຄຳຕອບ

Use the Quotient Property, $\frac{{a}^{m}}{{a}^{n}}={a}^{m-n}$.
Simplify.
Simplify: $\frac{{x}^{8}}{{x}^{-3}}.$

ເປີດເຜີຍຄຳຕອບ

x¹¹
Simplify: $\frac{{y}^{7}}{{y}^{-6}}.$

ເປີດເຜີຍຄຳຕອບ

y¹³

Write $37,000$ in scientific notation.

ເປີດເຜີຍຄຳຕອບ

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.
	$37,000=3.7\times {10}^{4}$

Write in scientific notation: $96,000.$

ເປີດເຜີຍຄຳຕອບ

9.6 × 10⁴
Write in scientific notation: $48,300.$

ເປີດເຜີຍຄຳຕອບ

4.83 × 10⁴

Write in scientific notation: $0.0052.$

ເປີດເຜີຍຄຳຕອບ

	0.0052
Move the decimal point to get 5.2, a number between 1 and 10.
Count the number of decimal places the point was moved.	3 places
Write as a product with a power of 10.	$5.2\times {10}^{-3}$
Check your answer: $\begin{array}{l}5.2\ \times \ {10}^{-3} \\ 5.2\ \times \ \frac{1}{{10}^{3}}\ \\ \\ \\ 5.2\ \times \ \frac{1}{1000} \\ 5.2\ \times \ 0.001 \\ 0.0052\end{array}$
	$0.0052=5.2\ \times \ {10}^{-3}$

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.

\approx

approximately equal
Equal to the precision shown, not exactly.

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

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

a \bmod n

remainder
What is left after dividing a by n.

per cent
Per hundred: 15% = 15/100.

a : b,\ \frac{a}{b}

ratio, fraction
a for every b; a divided by b.

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
The power of
The power of
greater than 1, the power of 10 will be

Questions people ask

Why does the order of operations matter?

Because 2 + 3 × 4 would otherwise be two different numbers. The convention (brackets, exponents, multiplication and division, addition and subtraction) exists so every reader gets the same value from the same expression.

How do I check an arithmetic answer?

Estimate first (round every number and compute roughly), then compare. If the estimate and the exact answer disagree by more than a little, one of them is wrong. The solver shows every operation, so you can find which line went astray.

Why are fractions harder than decimals?

They are not harder, they are more exact: 1/3 is a precise number, 0.333 is an approximation. Fractions need a common denominator to add, which is the one extra step people trip on.

ພະຍາຍາມເອງ

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

ເພີ່ມເຕີມໃນ Arithmetic

Order of operations Fractions Percentages Powers and roots

ⓐ
	\({4}^{-2}\)
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.	\(\frac{1}{1000}\)

ⓐ
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.	\(-\frac{1}{9}\)

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

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

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.	\(6.2\times {10}^{3}=6,200\)

	\(8.9\times {10}^{-2}\)
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.

	\((4\ \times \ {10}^{5})(2\ \times \ {10}^{-7})\)
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.	\(0.08\)

	\(\frac{9\ \times \ {10}^{3}}{3\ \times \ {10}^{-2}}\)
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.	\(300,000\)

ⓐ
Do exponents before multiplication.	\(4\cdot {2}^{-1}\)
Use \({a}^{-n}=\frac{1}{{a}^{n}}.\)	\(4\cdot \frac{1}{{2}^{1}}\)
Simplify.	\(2\)

ⓑ	\({(4\cdot 2)}^{-1}\)
Simplify inside the parentheses first.	\({(8)}^{-1}\)
Use \({a}^{-n}=\frac{1}{{a}^{n}}.\)	\(\frac{1}{{8}^{1}}\)
Simplify.	\(\frac{1}{8}\)

	\({x}^{-6}\)
Use the definition of a negative exponent, \({a}^{-n}=\frac{1}{{a}^{n}}.\)	\(\frac{1}{{x}^{6}}\)

ⓐ
Notice the exponent applies to just the base \(y\).	\(5{y}^{-1}\)
Take the reciprocal of \(y\) and change the sign of the exponent.	\(5\cdot \frac{1}{{y}^{1}}\)
Simplify.	\(\frac{5}{y}\)

ⓑ
Here the parentheses make the exponent apply to the base \(5y\).	\({(5y)}^{-1}\)
Take the reciprocal of \(5y\) and change the sign of the exponent.	\(\frac{1}{{(5y)}^{1}}\)
Simplify.	\(\frac{1}{5y}\)

ⓒ
	\({(-5y)}^{-1}\)
The base is \(-5y\). Take the reciprocal of \(-5y\) and change the sign of the exponent.	\(\frac{1}{{(-5y)}^{1}}\)
Simplify.	\(\frac{1}{-5y}\)
Use \(\frac{a}{-b}=-\frac{a}{b}.\)	\(-\frac{1}{5y}\)

	\((2{x}^{-6}{y}^{8})(-5{x}^{5}{y}^{-3})\)
Rewrite with the like bases together.	\(2(-5)\cdot ({x}^{-6}{x}^{5})\cdot ({y}^{8}{y}^{-3})\)
Simplify.	\(-10\cdot {x}^{-1}\cdot {y}^{5}\)
Use the definition of a negative exponent, \({a}^{-n}=\frac{1}{{a}^{n}}.\)	\(-10\cdot \frac{1}{{x}^{1}}\cdot {y}^{5}\)
Simplify.	\(\frac{-10{y}^{5}}{x}\)

	\({({k}^{3})}^{-2}\)
Use the Product to a Power Property, \({(ab)}^{m}={a}^{m}{b}^{m}.\)	\({k}^{3(-2)}\)
Simplify.	\({k}^{-6}\)
Rewrite with a positive exponent.	\(\frac{1}{{k}^{6}}\)

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


Use the Quotient Property, \(\frac{{a}^{m}}{{a}^{n}}={a}^{m-n}\).

Simplify.

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

Key Concepts

Integer Exponents and Scientific Notation

Practice (40)

Symbols used here

How to: Integer Exponents and Scientific Notation

Questions people ask

ພະຍາຍາມ​ເອງ

ເພີ່ມເຕີມໃນ Arithmetic

ພະຍາຍາມເອງ