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

 Page 8 / 9

## 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

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Mark Reply
nothing up todat yet
Miranda
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jai
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jai
Miranda Drice
jai
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jai
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Miranda
I am living in india
jai
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Miranda
what is the formula for calculating algebraic
Propessor Reply
I think the formula for calculating algebraic is the statement of the equality of two expression stimulate by a set of addition, multiplication, soustraction, division, raising to a power and extraction of Root. U believe by having those in the equation you will be in measure to calculate it
Miranda
state and prove Cayley hamilton therom
sita Reply
hello
Propessor
hi
Miranda
the Cayley hamilton Theorem state if A is a square matrix and if f(x) is its characterics polynomial then f(x)=0 in another ways evey square matrix is a root of its chatacteristics polynomial.
Miranda
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jai
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jai
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Propessor
welcome
jai
What is algebra
Pearl Reply
algebra is a branch of the mathematics to calculate expressions follow.
Miranda
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Jeffrey
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Miranda
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Miranda
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Jeffrey
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Miranda
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Miranda
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Jeffrey
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Jeffrey
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Miranda
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Miranda
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Steve
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Steve
I don't know why. But Im trying to like it.
Jeffrey
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Jeffrey
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Miranda
what is the solution of the given equation?
Nelson Reply
which equation
Miranda
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Jeffrey
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Miranda
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Jeffrey
answer and questions in exercise 11.2 sums
Yp Reply
how do u calculate inequality of irrational number?
Alaba
give me an example
Chris
and I will walk you through it
Chris
cos (-z)= cos z .
Swadesh
cos(- z)=cos z
Mustafa
what is a algebra
Jallah Reply
(x+x)3=?
Narad
6x
Obed
what is the identity of 1-cos²5x equal to?
liyemaikhaya Reply
__john __05
Kishu
Hi
Abdel
hi
Ye
hi
Nokwanda
C'est comment
Abdel
Hi
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hello
SORIE
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hello
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SORIE
it's 12
what is the function of sine with respect of cosine , graphically
Karl Reply
tangent bruh
Steve
cosx.cos2x.cos4x.cos8x
Aashish Reply
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

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