# 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? Answers will vary. ## 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? #### Questions & Answers are nano particles real Missy Reply yeah Joseph Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master? Lale Reply no can't Lohitha where we get a research paper on Nano chemistry....? Maira Reply nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review Ali what are the products of Nano chemistry? Maira Reply There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others.. learn Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level learn Google da no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts Bhagvanji hey Giriraj Preparation and Applications of Nanomaterial for Drug Delivery Hafiz Reply revolt da Application of nanotechnology in medicine has a lot of application modern world Kamaluddeen yes narayan what is variations in raman spectra for nanomaterials Jyoti Reply ya I also want to know the raman spectra Bhagvanji I only see partial conversation and what's the question here! Crow Reply what about nanotechnology for water purification RAW Reply please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment. Damian yes that's correct Professor I think Professor Nasa has use it in the 60's, copper as water purification in the moon travel. Alexandre nanocopper obvius Alexandre what is the stm Brian Reply is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.? Rafiq industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong Damian How we are making nano material? LITNING Reply what is a peer LITNING Reply What is meant by 'nano scale'? LITNING Reply What is STMs full form? LITNING scanning tunneling microscope Sahil how nano science is used for hydrophobicity Santosh Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq Rafiq what is differents between GO and RGO? Mahi what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq Rafiq if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION Anam analytical skills graphene is prepared to kill any type viruses . Anam Any one who tell me about Preparation and application of Nanomaterial for drug Delivery Hafiz what is Nano technology ? Bob Reply write examples of Nano molecule? Bob The nanotechnology is as new science, to scale nanometric brayan nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale Damian Is there any normative that regulates the use of silver nanoparticles? Damian Reply what king of growth are you checking .? Renato 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 A soccer field is a rectangle 130 meters wide and 110 meters long. The coach asks players to run from one corner to the other corner diagonally across. What is that distance, to the nearest tenths place. Kimberly Reply Jeannette has$5 and \$10 bills in her wallet. The number of fives is three more than six times the number of tens. Let t represent the number of tens. Write an expression for the number of fives.
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