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.

3. When Jenna spent 10 minutes on the elliptical trainer and then did circuit training for20 minutes, her fitness app says she burned 278 calories. When she spent 20 minutes onthe elliptical trainer and 30 minutes circuit training she burned 473 calories. How manycalories does she burn for each minute on the elliptical trainer? How many calories doesshe burn for each minute of circuit training?
John left his house in Irvine at 8:35 am to drive to a meeting in Los Angeles, 45 miles away. He arrived at the meeting at 9:50. At 3:30 pm, he left the meeting and drove home. He arrived home at 5:18.
p-2/3=5/6 how do I solve it with explanation pls
P=3/2
Vanarith
1/2p2-2/3p=5p/6
James
Cindy
4.5
Ruth
is y=7/5 a solution of 5y+3=10y-4
yes
James
Cindy
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Cindy
the sum of two Numbers is 19 and their difference is 15
2, 17
Jose
interesting
saw
4,2
Cindy
Felecia left her home to visit her daughter, driving 45mph. Her husband waited for the dog sitter to arrive and left home 20 minutes, or 13 hour later. He drove 55mph to catch up to Felecia. How long before he reaches her?
integer greater than 2 and less than 12
2 < x < 12
Felix
I'm guessing you are doing inequalities...
Felix
Actually, translating words into algebraic expressions / equations...
Felix
hi
Darianna
hello
Mister
Eric here
Eric
6
Cindy
He charges $125 per job. His monthly expenses are$1,600. How many jobs must he work in order to make a profit of at least \$2,400?
at least 20
Ayla
what are the steps?
Alicia
6.4 jobs
Grahame
32
Grahame
1600+2400= total amount with expenses. 4000/125= number of jobs needed to make that min profit of 2400. answer is 32
Orlando
He must work 32 jobs to make a profit
POP
32
Cindy
what is algebra
repeated addition and subtraction of the order of operations. i love algebra I'm obsessed.
Shemiah
hi
Krekar
Eric here. I'm a parent. 53 years old. I have never taken algebra. I want to learn.
Eric
I am 63 and never learned algebra
Cindy
One-fourth of the candies in a bag of M&M’s are red. If there are 23 red candies, how many candies are in the bag?
they are 92 candies in the bag
POP
92
Cindy