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: $6.2 \times 10^{3} .$

Solution

This figure is a table that has three columns and three rows. The first column is a header column, and it contains the names and numbers of each step. The second column contains further written instructions. The third column contains math. On the top row of the table, the first cell on the left reads “Step 1. Determine the exponent, n, on the factor 10.” The second cell reads “The exponent is 3.” The third cell contains 6.2 times 10 cubed.

In the second row, the first cell reads “Step 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 absolute value of n places to the left.” The second cell reads “The exponent is positive so move the decimal point 3 places to the right. We need to add two zeros as placeholders.” The third cell contains 6.200, with an arrow showing the decimal point jumping places to the right, from between the 6 and 2 to after the second 00 in 6.200. Below this is the number 6,200.

In the third row, the first cell reads “Step 3. Check to see if your answer makes sense.” The second cell is blank. The third reads “10 cubed is 1000 and 1000 times 6.2 will be 6,200.” Beneath this is 6.2 times 10 cubed equals 6,200.

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Convert to decimal form: $1.3 \times 10^{3} .$

1,300

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Convert to decimal form: $9.25 \times 10^{4} .$

92,500

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The steps are summarized below.

Convert scientific notation to decimal form.

To convert scientific notation to decimal form:

Determine the exponent, $n$ , on the factor 10.
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.
Check.

Convert to decimal form: $8.9 \times 10^{−2} .$

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.

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Convert to decimal form: $1.2 \times 10^{−4} .$

0.00012

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Convert to decimal form: $7.5 \times 10^{−2} .$

0.075

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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: $(4 \times 10^{5}) (2 \times 10^{−7}) .$

Solution

$\begin{matrix} (4 \times 10^{5}) (2 \times 10^{−7}) \\ Use the Commutative Property to rearrange the factors. & 4 \cdot 2 \cdot 10^{5} \cdot 10^{−7} \\ Multiply. & 8 \times 10^{−2} \\ Change to decimal form by moving the decimal two places left. & 0.08 \end{matrix}$

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Multiply $(3 \times 10^{6}) (2 \times 10^{−8})$ . Write answers in decimal form.

0.06

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Multiply $(3 \times 10^{−2}) (3 \times 10^{−1})$ . Write answers in decimal form.

0.009

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Divide. Write answers in decimal form: $\frac{9 \times 10^{3}}{3 \times 10^{−2}} .$

Solution

$\begin{matrix} \frac{9 \times 10^{3}}{3 \times 10^{−2}} \\ Separate the factors, rewriting as the product of two fractions. & \frac{9}{3} \times \frac{10^{3}}{10^{−2}} \\ Divide. & 3 \times 10^{5} \\ Change to decimal form by moving the decimal five places right. & 300,000 \end{matrix}$

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Divide $\frac{8 \times 10^{4}}{2 \times 10^{−1}}$ . Write answers in decimal form.

400,000

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Divide $\frac{8 \times 10^{2}}{4 \times 10^{−2}}$ . Write answers in decimal form.

20,000

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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 $a \neq 0$ , then $\frac{1}{a^{− n}} = a^{n}$
Quotient to a Negative Exponent
- If $a, b$ are real numbers, $b \neq 0$ and $n$ is an integer , then ${(\frac{a}{b})}^{− n} = {(\frac{b}{a})}^{n}$
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 $10^{− n}$
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 do you find divisible numbers without a calculator?

Jacob Reply

When she graduates college, Linda will owe $43,000 in student loans. The interest rate on the federal loans is 4.5% and the rate on the private bank loans is 2%. The total interest she owes for one year was $1,585. What is the amount of each loan?

Ariana Reply

Sean took the bus from Seattle to Boise, a distance of 506 miles. If the trip took 7 2/3 hours, what was the speed of the bus?

Kirisma Reply

66miles/hour

snigdha

How did you work it out?

Esther

s=mi/hr 2/3~0.67 s=506mi/7.67hr = ~66 mi/hr

Orlando

hello, I have algebra phobia. Subtracting negative numbers always seem to get me confused.

Alicia Reply

what do you need help in?

Felix

subtracting a negative....is adding!!

Heather

look at the numbers if they have different signs, it's like subtracting....but you keep the sign of the largest number...

Felix

for example.... -19 + 7.... different signs...subtract.... 12 keep the sign of the "largest" number 19 is bigger than 7.... 19 has the negative sign... Therefore, -12 is your answer...

Felix

—12

Niazmohammad

Thanks Felix.l also get confused with signs.

Esther

Thank you for this

Shatey

Graham

think about it like you lost $19 (-19), then found $7(+7). Totally you lost just $12 (-12)

Annushka

I used to struggle a lot with negative numbers and math in general what I typically do is look at it in terms of money I have -$5 in my account I then take out 5 more dollars how much do I have in my account well-$10 ... I also for a long time would draw it out on a number line to visualize it

Meg

practicing with smaller numbers to understand then working with larger numbers helps too and the song/rhyme same sign add and keep opposite signs subtract keep the sign of the bigger # then you'll be exact

Meg

Bruce drives his car for his job. The equation R=0.575m+42 models the relation between the amount in dollars, R, that he is reimbursed and the number of miles, m, he drives in one day. Find the amount Bruce is reimbursed on a day when he drives 220 miles

John Reply

168.50=R

Heather

john is 5years older than wanjiru.the sum of their years is27years.what is the age of each

achol Reply

mustee

j 17 w 11

Joseph

john is 16. wanjiru is 11.

Felix

27-5=22 22÷2=11 11+5=16

Joyce

where's the answers?

Ed Reply

I don't see where the answers are.

Cindy and Richard leave their dorm in Charleston at the same time. Cindy rides her bicycle north at a speed of 18 miles per hour. Richard rides his bicycle south at a speed of 14 miles per hour. How long will it take them to be 96 miles apart?

Maddy Reply

Christopher

18t+14t=96 32t=96 32/96 3

Christopher

show that a^n-b^2n is divisible by a-b

Florence Reply

What does 3 times your weight right now

Cherokee Reply

Use algebra to combine 39×5 and the half sum of travel of 59+30

Cherokee

What is the segment of 13? Explain

Cherokee

my weight is 49. So 3 times is 147

Cherokee

kg to lbs you goin to convert 2.2 or one if the same unit your going to time your body weight by 3. example if my body weight is 210lb. what would be my weight if I was 3 times as much in kg. that's you do 210 x3 = 630lb. then 630 x 2.2= .... hope this helps

tyler

How to convert grams to pounds?

paul

What is the lcm of 340

Kendra Reply

Yes

Cherokee

How many numbers each equal to y must be taken to make 15xy

Malik Reply

15x

Martin

15x

Asamoah

15x

Hugo

Tom

1y x 15y

Tom

find the equation whose roots are 1 and 2

Adda Reply

(x - 2)(x -1)=0 so equation is x^2-x+2=0

Ranu

I believe it's x^2-3x+2

NerdNamedGerg

because the X's multiply by the -2 and the -1 and than combine like terms

NerdNamedGerg

find the equation whose roots are -1 and 4

Adda

Ans = ×^2-3×+2

Gee

find the equation whose roots are -2 and -1

Adda

(×+1)(×-4) = x^2-3×-4

Gee

yeah

Asamoah

Quadratic equations involving factorization

Winner Reply

there's a chatting option in the app wow

Nana

That's cool cool

Nana

Nice to meet you all

Nana

you too.

Joan

😃

Nana

Hey you all there are several Free Apps that can really help you to better solve type Equations.

Debra

Debra, which apps specifically. ..?

Nana

am having a course in elementary algebra ,any recommendations ?

samuel

Samuel Addai, me too at ucc elementary algebra as part of my core subjects in science

Nana

me too as part of my core subjects in R M E

Ken

at ABETIFI COLLEGE OF EDUCATION

Ken

ok great. Good to know.

Joan

5x + 1/3= 2x + 1/2

sanam

Plz solve this

sanam

5x - 3x = 1/2 - 1/3 2x = 1/6 x = 1/12

Ranu

Thks ranu

sanam

Please help me solve this 10x+14=-2x+38

Erica

the previous equation should be 3x = 1/6 x=1/18

Sriram

for the new one 10x + 2x = 38 - 14

Sriram

12x = 24 x=2

Sriram

10x + 14 = -2x +38 10x + 2x = 38 - 14 12x = 24 divide both sides by the coefficient of x, which is 12 therefore × = 2

vida

a trader gains 20 rupees loses 42 rupees and then gains ten rupees Express algebraically the result of his transactions

vinaya Reply

a trader gains 20 rupees loses 42 rupees and then gains 10 rupees Express algebraically the result of his three transactions

vinaya

a trader gains 20 rupees loses 42 rupees and then gains 10 rupees Express algebraically the result of his three transactions

vinaya

a trader gains 20 rupees loses 42 rupees and then gains 10 rupees Express algebraically the result of his three transactions

vinaya

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Source: OpenStax, Elementary algebra. OpenStax CNX. Jan 18, 2017 Download for free at http://cnx.org/content/col12116/1.2

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