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

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## 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.

how did you get the value of 2000N.What calculations are needed to arrive at it
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