# 6.7 Integer exponents and scientific notation  (Page 5/10)

 Page 5 / 10

## How to convert scientific notation to decimal form

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

## Solution

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

1,300

Convert to decimal form: $9.25\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{4}.$

92,500

The steps are summarized below.

## 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.

Convert to decimal form: $8.9\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-2}.$

## 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.

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

0.00012

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

0.075

## 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.

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

## Solution

$\begin{array}{cccc}& & & \left(4\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{5}\right)\left(2\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-7}\right)\hfill \\ \\ \\ \text{Use the Commutative Property to rearrange the factors.}\hfill & & & 4·2·{10}^{5}·{10}^{-7}\hfill \\ \\ \\ \text{Multiply.}\hfill & & & 8\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-2}\hfill \\ \\ \\ \text{Change to decimal form by moving the decimal two places left.}\hfill & & & 0.08\hfill \end{array}$

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

0.06

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

0.009

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

## Solution

$\begin{array}{cccc}& & & \frac{9\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{3}}{3\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-2}}\hfill \\ \\ \\ \text{Separate the factors, rewriting as the product of two fractions.}\hfill & & & \frac{9}{3}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\frac{{10}^{3}}{{10}^{-2}}\hfill \\ \\ \\ \text{Divide.}\hfill & & & 3\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{5}\hfill \\ \\ \\ \text{Change to decimal form by moving the decimal five places right.}\hfill & & & 300,000\hfill \end{array}$

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

400,000

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

20,000

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

## 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 ${\left(\frac{a}{b}\right)}^{\text{−}n}={\left(\frac{b}{a}\right)}^{n}$
• To convert a decimal to scientific notation:
1. Move the decimal point so that the first factor is greater than or equal to 1 but less than 10.
2. Count the number of decimal places, $n$ , that the decimal point was moved.
3. Write the number as a product with a power of 10. If the original number is:
• greater than 1, the power of 10 will be ${10}^{n}$
• between 0 and 1, the power of 10 will be ${10}^{\text{−}n}$
4. Check.

• To convert scientific notation to decimal form:
1. Determine the exponent, $n$ , on the factor 10.
2. Move the decimal $n$ places, adding zeros if needed.
• If the exponent is positive, move the decimal point $n$ places to the right.
• If the exponent is negative, move the decimal point $|n|$ places to the left.
3. Check.

#### Questions & Answers

how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
Good
Two sisters like to compete on their bike rides. Tamara can go 4 mph faster than her sister, Samantha. If it takes Samantha 1 hours longer than Tamara to go 80 miles, how fast can Samantha ride her bike? Got questions? Get instant answers now!
Seera Reply
how do u solve that question
Seera
Two sisters like to compete on their bike rides. Tamara can go 4 mph faster than her sister, Samantha. If it takes Samantha 1 hours longer than Tamara to go 80 miles, how fast can Samantha ride her bike?
Seera
Speed=distance ÷ time
Tremayne
x-3y =1; 3x-2y+4=0 graph
Juned Reply
Brandon has a cup of quarters and dimes with a total of 5.55$. The number of quarters is five less than three times the number of dimes ashley Reply app is wrong how can 350 be divisible by 3. Raheem Reply June needs 48 gallons of punch for a party and has two different coolers to carry it in. The bigger cooler is five times as large as the smaller cooler. How many gallons can each cooler hold? Susanna Reply Susanna if the first cooler holds five times the gallons then the other cooler. The big cooler holda 40 gallons and the 2nd will hold 8 gallons is that correct? Georgie @Susanna that person is correct if you divide 40 by 8 you can see it's 5 it's simple Ashley @Geogie my bad that was meant for u Ashley Hi everyone, I'm glad to be connected with you all. from France. Lorris Reply I'm getting "math processing error" on math problems. Anyone know why? Ray Reply Can you all help me I don't get any of this Jade Reply 4^×=9 Alberto Reply Did anyone else have trouble getting in quiz link for linear inequalities? Sireka Reply operation of trinomial Justin Reply y=2×+9 Jacob Reply Keshad gets paid$2,400 per month plus 6% of his sales. His brother earns $3,300 per month. For what amount of total sales will Keshad’s monthly pay be higher than his brother’s monthly pay? Hector Reply Mayra has$124 in her checking account. She writes a check for $152. What is the New Balance in her checking account? REVOLUTION Reply -28$
ashley
-\$28
Stephanie

### Read also:

#### Get the best Elementary algebra course in your pocket!

Source:  OpenStax, Elementary algebra. OpenStax CNX. Jan 18, 2017 Download for free at http://cnx.org/content/col12116/1.2
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Elementary algebra' conversation and receive update notifications?

 By Qqq Qqq By By By