# 10.5 Integer exponents and scientific notation  (Page 5/7)

 Page 5 / 7

## Key concepts

• Summary of Exponent Properties
• If $a,b$ are real numbers and $m,n$ are integers, then
$\begin{array}{cccc}\mathbf{\text{Product Property}}\hfill & & & {a}^{m}·{a}^{n}={a}^{m+n}\hfill \\ \mathbf{\text{Power Property}}\hfill & & & {\left({a}^{m}\right)}^{n}={a}^{m·n}\hfill \\ \mathbf{\text{Product to a Power Property}}\hfill & & & {\left(ab\right)}^{m}={a}^{m}{b}^{m}\hfill \\ \mathbf{\text{Quotient Property}}\hfill & & & \frac{{a}^{m}}{{a}^{n}}={a}^{m-n},\phantom{\rule{0.2em}{0ex}}a\ne 0\hfill \\ \mathbf{\text{Zero Exponent Property}}\hfill & & & {a}^{0}=1,\phantom{\rule{0.2em}{0ex}}a\ne 0\hfill \\ \mathbf{\text{Quotient to a Power Property}}\hfill & & & {\left(\frac{a}{b}\right)}^{m}=\frac{{a}^{m}}{{b}^{m}},\phantom{\rule{0.2em}{0ex}}b\ne 0\hfill \\ \mathbf{\text{Definition of Negative Exponent}}\hfill & & & {a}^{-n}=\frac{1}{{a}^{n}}\hfill \end{array}$
• 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.

## Practice makes perfect

Use the Definition of a Negative Exponent

In the following exercises, simplify.

${5}^{-3}$

${8}^{-2}$

$\frac{1}{64}$

${3}^{-4}$

${2}^{-5}$

$\frac{1}{32}$

${7}^{-1}$

${10}^{-1}$

$\frac{1}{10}$

${2}^{-3}+{2}^{-2}$

${3}^{-2}+{3}^{-1}$

$\frac{4}{9}$

${3}^{-1}+{4}^{-1}$

${10}^{-1}+{2}^{-1}$

$\frac{3}{5}$

${10}^{0}-{10}^{-1}+{10}^{-2}$

${2}^{0}-{2}^{-1}+{2}^{-2}$

$\frac{3}{4}$

1. $\phantom{\rule{0.2em}{0ex}}{\left(-6\right)}^{-2}$
2. $\phantom{\rule{0.2em}{0ex}}-{6}^{-2}$

1. $\phantom{\rule{0.2em}{0ex}}{\left(-8\right)}^{-2}$
2. $\phantom{\rule{0.2em}{0ex}}-{8}^{-2}$

1. $\phantom{\rule{0.2em}{0ex}}\frac{1}{64}$
2. $\phantom{\rule{0.2em}{0ex}}-\frac{1}{64}$

1. $\phantom{\rule{0.2em}{0ex}}{\left(-10\right)}^{-4}$
2. $\phantom{\rule{0.2em}{0ex}}-{10}^{-4}$

1. $\phantom{\rule{0.2em}{0ex}}{\left(-4\right)}^{-6}$
2. $\phantom{\rule{0.2em}{0ex}}-{4}^{-6}$

1. $\phantom{\rule{0.2em}{0ex}}\frac{1}{4096}$
2. $\phantom{\rule{0.2em}{0ex}}-\frac{1}{4096}$

1. $\phantom{\rule{0.2em}{0ex}}5·{2}^{-1}$
2. $\phantom{\rule{0.2em}{0ex}}{\left(5·2\right)}^{-1}$

1. $\phantom{\rule{0.2em}{0ex}}10·{3}^{-1}$
2. $\phantom{\rule{0.2em}{0ex}}{\left(10·3\right)}^{-1}$

1. $\phantom{\rule{0.2em}{0ex}}\frac{10}{3}$
2. $\phantom{\rule{0.2em}{0ex}}\frac{1}{30}$

1. $\phantom{\rule{0.2em}{0ex}}4·{10}^{-3}$
2. $\phantom{\rule{0.2em}{0ex}}{\left(4·10\right)}^{-3}$

1. $\phantom{\rule{0.2em}{0ex}}3·{5}^{-2}$
2. $\phantom{\rule{0.2em}{0ex}}{\left(3·5\right)}^{-2}$

1. $\phantom{\rule{0.2em}{0ex}}\frac{3}{25}$
2. $\phantom{\rule{0.2em}{0ex}}\frac{1}{225}$

${n}^{-4}$

${p}^{-3}$

$\frac{1}{{p}^{3}}$

${c}^{-10}$

${m}^{-5}$

$\frac{1}{{m}^{5}}$

1. $\phantom{\rule{0.2em}{0ex}}4{x}^{-1}$
2. $\phantom{\rule{0.2em}{0ex}}{\left(4x\right)}^{-1}$
3. $\phantom{\rule{0.2em}{0ex}}{\left(-4x\right)}^{-1}$

1. $\phantom{\rule{0.2em}{0ex}}3{q}^{-1}$
2. $\phantom{\rule{0.2em}{0ex}}{\left(3q\right)}^{-1}$
3. $\phantom{\rule{0.2em}{0ex}}{\left(-3q\right)}^{-1}$

1. $\phantom{\rule{0.2em}{0ex}}\frac{3}{q}$
2. $\phantom{\rule{0.2em}{0ex}}\frac{1}{3q}$
3. $\phantom{\rule{0.2em}{0ex}}-\frac{1}{3q}$

1. $\phantom{\rule{0.2em}{0ex}}6{m}^{-1}$
2. $\phantom{\rule{0.2em}{0ex}}{\left(6m\right)}^{-1}$
3. $\phantom{\rule{0.2em}{0ex}}{\left(-6m\right)}^{-1}$

1. $\phantom{\rule{0.2em}{0ex}}10{k}^{-1}$
2. $\phantom{\rule{0.2em}{0ex}}{\left(10k\right)}^{-1}$
3. $\phantom{\rule{0.2em}{0ex}}{\left(-10k\right)}^{-1}$

1. $\phantom{\rule{0.2em}{0ex}}\frac{10}{k}$
2. $\phantom{\rule{0.2em}{0ex}}\frac{1}{10k}$
3. $\phantom{\rule{0.2em}{0ex}}-\frac{1}{10k}$

Simplify Expressions with Integer Exponents

In the following exercises, simplify .

${p}^{-4}·{p}^{8}$

${r}^{-2}·{r}^{5}$

r 3

${n}^{-10}·{n}^{2}$

${q}^{-8}·{q}^{3}$

$\frac{1}{{q}^{5}}$

${k}^{-3}·{k}^{-2}$

${z}^{-6}·{z}^{-2}$

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

$a·{a}^{-4}$

$m·{m}^{-2}$

$\frac{1}{m}$

${p}^{5}·{p}^{-2}·{p}^{-4}$

${x}^{4}·{x}^{-2}·{x}^{-3}$

$\frac{1}{x}$

${a}^{3}{b}^{-3}$

${u}^{2}{v}^{-2}$

$\frac{{u}^{2}}{{v}^{2}}$

$\left({x}^{5}{y}^{-1}\right)\left({x}^{-10}{y}^{-3}\right)$

$\left({a}^{3}{b}^{-3}\right)\left({a}^{-5}{b}^{-1}\right)$

$\frac{1}{{a}^{2}{b}^{4}}$

$\left(u{v}^{-2}\right)\left({u}^{-5}{v}^{-4}\right)$

$\left(p{q}^{-4}\right)\left({p}^{-6}{q}^{-3}\right)$

$\frac{1}{{p}^{5}{q}^{7}}$

$\left(-2{r}^{-3}{s}^{9}\right)\left(6{r}^{4}{s}^{-5}\right)$

$\left(-3{p}^{-5}{q}^{8}\right)\left(7{p}^{2}{q}^{-3}\right)$

$-\frac{21{q}^{5}}{{p}^{3}}$

$\left(-6{m}^{-8}{n}^{-5}\right)\left(-9{m}^{4}{n}^{2}\right)$

$\left(-8{a}^{-5}{b}^{-4}\right)\left(-4{a}^{2}{b}^{3}\right)$

$\frac{32}{{a}^{3}b}$

${\left({a}^{3}\right)}^{-3}$

${\left({q}^{10}\right)}^{-10}$

$\frac{1}{{q}^{100}}$

${\left({n}^{2}\right)}^{-1}$

${\left({x}^{4}\right)}^{-1}$

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

${\left({y}^{-5}\right)}^{4}$

${\left({p}^{-3}\right)}^{2}$

$\frac{1}{{y}^{6}}$

${\left({q}^{-5}\right)}^{-2}$

${\left({m}^{-2}\right)}^{-3}$

m 6

${\left(4{y}^{-3}\right)}^{2}$

${\left(3{q}^{-5}\right)}^{2}$

$\frac{9}{{q}^{10}}$

${\left(10{p}^{-2}\right)}^{-5}$

${\left(2{n}^{-3}\right)}^{-6}$

$\frac{{n}^{18}}{64}$

$\frac{{u}^{9}}{{u}^{-2}}$

$\frac{{b}^{5}}{{b}^{-3}}$

b 8

$\frac{{x}^{-6}}{{x}^{4}}$

$\frac{{m}^{5}}{{m}^{-2}}$

m 7

$\frac{{q}^{3}}{{q}^{12}}$

$\frac{{r}^{6}}{{r}^{9}}$

$\frac{1}{{r}^{3}}$

$\frac{{n}^{-4}}{{n}^{-10}}$

$\frac{{p}^{-3}}{{p}^{-6}}$

p 3

Convert from Decimal Notation to Scientific Notation

In the following exercises, write each number in scientific notation.

45,000

280,000

2.8 × 10 5

8,750,000

1,290,000

1.29 × 10 6

0.036

0.041

4.1 × 10 −2

0.00000924

0.0000103

1.03 × 10 −5

The population of the United States on July 4, 2010 was almost $310,000,000.$

The population of the world on July 4, 2010 was more than $6,850,000,000.$

6.85 × 10 9

The average width of a human hair is $0.0018$ centimeters.

The probability of winning the $2010$ Megamillions lottery is about $0.0000000057.$

5.7 × 10 −9

Convert Scientific Notation to Decimal Form

In the following exercises, convert each number to decimal form.

$4.1\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{2}$

$8.3\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{2}$

830

$5.5\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{8}$

$1.6\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{10}$

16,000,000,000

$3.5\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-2}$

$2.8\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-2}$

0.028

$1.93\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-5}$

$6.15\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-8}$

0.0000000615

In 2010, the number of Facebook users each day who changed their status to ‘engaged’ was $2\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{4}.$

At the start of 2012, the US federal budget had a deficit of more than $\text{1.5}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{13}.$

\$15,000,000,000,000

The concentration of carbon dioxide in the atmosphere is $3.9\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-4}.$

The width of a proton is $1\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-5}$ of the width of an atom.

0.00001

Multiply and Divide Using Scientific Notation

In the following exercises, multiply or divide and write your answer in decimal form.

$\left(2\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{5}\right)\left(2\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-9}\right)$

$\left(3\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{2}\right)\left(1\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-5}\right)$

0.003

$\left(1.6\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-2}\right)\left(5.2\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-6}\right)$

$\left(2.1\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-4}\right)\left(3.5\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-2}\right)$

0.00000735

$\frac{6\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{4}}{3\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-2}}$

$\frac{8\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{6}}{4\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-1}}$

200,000

$\frac{7\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-2}}{1\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-8}}$

$\frac{5\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-3}}{1\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-10}}$

50,000,000

## Everyday math

Calories In May 2010 the Food and Beverage Manufacturers pledged to reduce their products by $1.5$ trillion calories by the end of 2015.

1. Write $1.5$ trillion in decimal notation.
2. Write $1.5$ trillion in scientific notation.

Length of a year The difference between the calendar year and the astronomical year is $0.000125$ day.

1. Write this number in scientific notation.
2. How many years does it take for the difference to become 1 day?
1. 1.25 × 10 −4
2. 8,000

Calculator display Many calculators automatically show answers in scientific notation if there are more digits than can fit in the calculator’s display. To find the probability of getting a particular 5-card hand from a deck of cards, Mario divided $1$ by $2,598,960$ and saw the answer $3.848\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-7}.$ Write the number in decimal notation.

Calculator display Many calculators automatically show answers in scientific notation if there are more digits than can fit in the calculator’s display. To find the number of ways Barbara could make a collage with $6$ of her $50$ favorite photographs, she multiplied $50·49·48·47·46·45.$ Her calculator gave the answer $1.1441304\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{10}.$ Write the number in decimal notation.

11,441,304,000

## Writing exercises

1. Explain the meaning of the exponent in the expression ${2}^{3}.$
2. Explain the meaning of the exponent in the expression ${2}^{-3}$

When you convert a number from decimal notation to scientific notation, how do you know if the exponent will be positive or negative?

## Self check

After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

After looking at the checklist, do you think you are well prepared for the next section? Why or why not?

can someone help me with some logarithmic and exponential equations.
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
I don't understand what the A with approx sign and the boxed x mean
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
oops. ignore that.
so you not have an equal sign anywhere in the original equation?
Commplementary angles
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
a perfect square v²+2v+_
kkk nice
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
y=10×
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
rolling four fair dice and getting an even number an all four dice
Kristine 2*2*2=8
Differences Between Laspeyres and Paasche Indices
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
is it 3×y ?
J, combine like terms 7x-4y
im not good at math so would this help me
yes
Asali
I'm not good at math so would you help me
Samantha
what is the problem that i will help you to self with?
Asali
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
what is nanomaterials​ and their applications of sensors.
what is nano technology
what is system testing?
preparation of nanomaterial
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
what is system testing
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
can nanotechnology change the direction of the face of the world
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
how do you translate this in Algebraic Expressions
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?