# 1.2 Exponents and scientific notation  (Page 8/9)

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## Key equations

 Rules of Exponents For nonzero real numbers $\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}b\text{\hspace{0.17em}}$ and integers $\text{\hspace{0.17em}}m\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}n\text{\hspace{0.17em}}$ Product rule ${a}^{m}\cdot {a}^{n}={a}^{m+n}$ Quotient rule $\frac{{a}^{m}}{{a}^{n}}={a}^{m-n}$ Power rule ${\left({a}^{m}\right)}^{n}={a}^{m\cdot n}$ Zero exponent rule ${a}^{0}=1$ Negative rule ${a}^{-n}=\frac{1}{{a}^{n}}$ Power of a product rule ${\left(a\cdot b\right)}^{n}={a}^{n}\cdot {b}^{n}$ Power of a quotient rule ${\left(\frac{a}{b}\right)}^{n}=\frac{{a}^{n}}{{b}^{n}}$

## Key concepts

• Products of exponential expressions with the same base can be simplified by adding exponents. See [link] .
• Quotients of exponential expressions with the same base can be simplified by subtracting exponents. See [link] .
• Powers of exponential expressions with the same base can be simplified by multiplying exponents. See [link] .
• An expression with exponent zero is defined as 1. See [link] .
• An expression with a negative exponent is defined as a reciprocal. See [link] and [link] .
• The power of a product of factors is the same as the product of the powers of the same factors. See [link] .
• The power of a quotient of factors is the same as the quotient of the powers of the same factors. See [link] .
• The rules for exponential expressions can be combined to simplify more complicated expressions. See [link] .
• Scientific notation uses powers of 10 to simplify very large or very small numbers. See [link] and [link] .
• Scientific notation may be used to simplify calculations with very large or very small numbers. See [link] and [link] .

## Verbal

Is $\text{\hspace{0.17em}}{2}^{3}\text{\hspace{0.17em}}$ the same as $\text{\hspace{0.17em}}{3}^{2}?\text{\hspace{0.17em}}$ Explain.

No, the two expressions are not the same. An exponent tells how many times you multiply the base. So $\text{\hspace{0.17em}}{2}^{3}\text{\hspace{0.17em}}$ is the same as $\text{\hspace{0.17em}}2×2×2,$ which is 8. $\text{\hspace{0.17em}}{3}^{2}\text{\hspace{0.17em}}$ is the same as $\text{\hspace{0.17em}}3×3,$ which is 9.

When can you add two exponents?

What is the purpose of scientific notation?

It is a method of writing very small and very large numbers.

Explain what a negative exponent does.

## Numeric

For the following exercises, simplify the given expression. Write answers with positive exponents.

$\text{\hspace{0.17em}}{9}^{2}\text{\hspace{0.17em}}$

81

${15}^{-2}$

${3}^{2}\text{\hspace{0.17em}}×\text{\hspace{0.17em}}{3}^{3}$

243

${4}^{4}÷4$

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

$\frac{1}{16}$

${\left(5-8\right)}^{0}$

${11}^{3}÷{11}^{4}$

$\frac{1}{11}$

${6}^{5}\text{\hspace{0.17em}}×\text{\hspace{0.17em}}{6}^{-7}$

${\left({8}^{0}\right)}^{2}$

1

${5}^{-2}÷{5}^{2}$

For the following exercises, write each expression with a single base. Do not simplify further. Write answers with positive exponents.

${4}^{2}\text{\hspace{0.17em}}×\text{\hspace{0.17em}}{4}^{3}÷{4}^{-4}$

${4}^{9}$

$\frac{{6}^{12}}{{6}^{9}}$

${\left({12}^{3}\text{\hspace{0.17em}}×\text{\hspace{0.17em}}12\right)}^{10}$

${12}^{40}$

${10}^{6}÷{\left({10}^{10}\right)}^{-2}$

${7}^{-6}\text{\hspace{0.17em}}×\text{\hspace{0.17em}}{7}^{-3}$

$\frac{1}{{7}^{9}}$

${\left({3}^{3}÷{3}^{4}\right)}^{5}$

For the following exercises, express the decimal in scientific notation.

0.0000314

$3.14\text{\hspace{0.17em}}×{10}^{-5}$

148,000,000

For the following exercises, convert each number in scientific notation to standard notation.

$1.6\text{\hspace{0.17em}}×\text{\hspace{0.17em}}{10}^{10}$

16,000,000,000

$9.8\text{\hspace{0.17em}}×\text{\hspace{0.17em}}{10}^{-9}$

## Algebraic

For the following exercises, simplify the given expression. Write answers with positive exponents.

$\frac{{a}^{3}{a}^{2}}{a}$

${a}^{4}$

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

${\left({b}^{3}{c}^{4}\right)}^{2}$

${b}^{6}{c}^{8}$

${\left(\frac{{x}^{-3}}{{y}^{2}}\right)}^{-5}$

$a{b}^{2}÷{d}^{-3}$

$a{b}^{2}{d}^{3}$

${\left({w}^{0}{x}^{5}\right)}^{-1}$

$\frac{{m}^{4}}{{n}^{0}}$

${m}^{4}$

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

$\frac{{p}^{-4}{q}^{2}}{{p}^{2}{q}^{-3}}$

$\frac{{q}^{5}}{{p}^{6}}$

${\left(l\text{\hspace{0.17em}}×\text{\hspace{0.17em}}w\right)}^{2}$

${\left({y}^{7}\right)}^{3}÷{x}^{14}$

$\frac{{y}^{21}}{{x}^{14}}$

${\left(\frac{a}{{2}^{3}}\right)}^{2}$

${5}^{2}m÷{5}^{0}m$

$25$

$\frac{{\left(16\sqrt{x}\right)}^{2}}{{y}^{-1}}$

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

$72{a}^{2}$

${\left(m{a}^{6}\right)}^{2}\frac{1}{{m}^{3}{a}^{2}}$

${\left({b}^{-3}c\right)}^{3}$

$\frac{{c}^{3}}{{b}^{9}}$

${\left({x}^{2}{y}^{13}÷{y}^{0}\right)}^{2}$

${\left(9{z}^{3}\right)}^{-2}y$

$\frac{y}{81{z}^{6}}$

## Real-world applications

To reach escape velocity, a rocket must travel at the rate of $\text{\hspace{0.17em}}2.2\text{\hspace{0.17em}}×\text{\hspace{0.17em}}{10}^{6}\text{\hspace{0.17em}}$ ft/min. Rewrite the rate in standard notation.

A dime is the thinnest coin in U.S. currency. A dime’s thickness measures $\text{\hspace{0.17em}}1.35\text{\hspace{0.17em}}×\text{\hspace{0.17em}}{10}^{-3}\text{\hspace{0.17em}}$ m. Rewrite the number in standard notation.

0.00135 m

The average distance between Earth and the Sun is 92,960,000 mi. Rewrite the distance using scientific notation.

A terabyte is made of approximately 1,099,500,000,000 bytes. Rewrite in scientific notation.

$1.0995×{10}^{12}$

The Gross Domestic Product (GDP) for the United States in the first quarter of 2014 was $\text{\hspace{0.17em}}\text{}1.71496\text{\hspace{0.17em}}×\text{\hspace{0.17em}}{10}^{13}.\text{\hspace{0.17em}}$ Rewrite the GDP in standard notation.

One picometer is approximately $\text{\hspace{0.17em}}3.397\text{\hspace{0.17em}}×\text{\hspace{0.17em}}{10}^{-11}\text{\hspace{0.17em}}$ in. Rewrite this length using standard notation.

0.00000000003397 in.

The value of the services sector of the U.S. economy in the first quarter of 2012 was \$10,633.6 billion. Rewrite this amount in scientific notation.

## Technology

For the following exercises, use a graphing calculator to simplify. Round the answers to the nearest hundredth.

${\left(\frac{{12}^{3}{m}^{33}}{{4}^{-3}}\right)}^{2}$

12,230,590,464 $\text{\hspace{0.17em}}{m}^{66}$

${17}^{3}÷{15}^{2}{x}^{3}$

## Extensions

For the following exercises, simplify the given expression. Write answers with positive exponents.

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

$\frac{{a}^{14}}{1296}$

${\left({6}^{2}-24\right)}^{2}÷{\left(\frac{x}{y}\right)}^{-5}$

$\frac{{m}^{2}{n}^{3}}{{a}^{2}{c}^{-3}}\cdot \frac{{a}^{-7}{n}^{-2}}{{m}^{2}{c}^{4}}$

$\frac{n}{{a}^{9}c}$

${\left(\frac{{x}^{6}{y}^{3}}{{x}^{3}{y}^{-3}}\cdot \frac{{y}^{-7}}{{x}^{-3}}\right)}^{10}$

${\left(\frac{{\left(a{b}^{2}c\right)}^{-3}}{{b}^{-3}}\right)}^{2}$

$\frac{1}{{a}^{6}{b}^{6}{c}^{6}}$

Avogadro’s constant is used to calculate the number of particles in a mole. A mole is a basic unit in chemistry to measure the amount of a substance. The constant is $\text{\hspace{0.17em}}6.0221413\text{\hspace{0.17em}}×\text{\hspace{0.17em}}{10}^{23}.\text{\hspace{0.17em}}$ Write Avogadro’s constant in standard notation.

Planck’s constant is an important unit of measure in quantum physics. It describes the relationship between energy and frequency. The constant is written as $\text{\hspace{0.17em}}6.62606957\text{\hspace{0.17em}}×\text{\hspace{0.17em}}{10}^{-34}.\text{\hspace{0.17em}}$ Write Planck’s constant in standard notation.

0.000000000000000000000000000000000662606957

#### Questions & Answers

sinx sin2x is linearly dependent
cr Reply
what is a reciprocal
Ajibola Reply
The reciprocal of a number is 1 divided by a number. eg the reciprocal of 10 is 1/10 which is 0.1
Shemmy
Reciprocal is a pair of numbers that, when multiplied together, equal to 1. Example; the reciprocal of 3 is ⅓, because 3 multiplied by ⅓ is equal to 1
Jeza
each term in a sequence below is five times the previous term what is the eighth term in the sequence
Funmilola Reply
I don't understand how radicals works pls
Kenny Reply
How look for the general solution of a trig function
collins Reply
stock therom F=(x2+y2) i-2xy J jaha x=a y=o y=b
Saurabh Reply
sinx sin2x is linearly dependent
cr
root under 3-root under 2 by 5 y square
Himanshu Reply
The sum of the first n terms of a certain series is 2^n-1, Show that , this series is Geometric and Find the formula of the n^th
amani Reply
cosA\1+sinA=secA-tanA
Aasik Reply
Wrong question
Saad
why two x + seven is equal to nineteen.
Kingsley Reply
The numbers cannot be combined with the x
Othman
2x + 7 =19
humberto
2x +7=19. 2x=19 - 7 2x=12 x=6
Yvonne
because x is 6
SAIDI
what is the best practice that will address the issue on this topic? anyone who can help me. i'm working on my action research.
Melanie Reply
simplify each radical by removing as many factors as possible (a) √75
Jason Reply
how is infinity bidder from undefined?
Karl Reply
what is the value of x in 4x-2+3
Vishal Reply
give the complete question
Shanky
4x=3-2 4x=1 x=1+4 x=5 5x
Olaiya
hi can you give another equation I'd like to solve it
Daniel
what is the value of x in 4x-2+3
Olaiya
if 4x-2+3 = 0 then 4x = 2-3 4x = -1 x = -(1÷4) is the answer.
Jacob
4x-2+3 4x=-3+2 4×=-1 4×/4=-1/4
LUTHO
then x=-1/4
LUTHO
4x-2+3 4x=-3+2 4x=-1 4x÷4=-1÷4 x=-1÷4
LUTHO
A research student is working with a culture of bacteria that doubles in size every twenty minutes. The initial population count was  1350  bacteria. Rounding to five significant digits, write an exponential equation representing this situation. To the nearest whole number, what is the population size after  3  hours?
David Reply
f(x)= 1350. 2^(t/20); where t is in hours.
Merkeb

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