# 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

The sequence is {1,-1,1-1.....} has
amit Reply
circular region of radious
Kainat Reply
how can we solve this problem
Joel Reply
Sin(A+B) = sinBcosA+cosBsinA
Eseka Reply
Prove it
Eseka
Please prove it
Eseka
hi
Joel
June needs 45 gallons of punch. 2 different coolers. Bigger cooler is 5 times as large as smaller cooler. How many gallons in each cooler?
Arleathia Reply
find the sum of 28th term of the AP 3+10+17+---------
Prince Reply
I think you should say "28 terms" instead of "28th term"
Vedant
if sequence sn is a such that sn>0 for all n and lim sn=0than prove that lim (s1 s2............ sn) ke hole power n =n
SANDESH Reply
write down the polynomial function with root 1/3,2,-3 with solution
Gift Reply
if A and B are subspaces of V prove that (A+B)/B=A/(A-B)
Pream Reply
write down the value of each of the following in surd form a)cos(-65°) b)sin(-180°)c)tan(225°)d)tan(135°)
Oroke Reply
Prove that (sinA/1-cosA - 1-cosA/sinA) (cosA/1-sinA - 1-sinA/cosA) = 4
kiruba Reply
what is the answer to dividing negative index
Morosi Reply
In a triangle ABC prove that. (b+c)cosA+(c+a)cosB+(a+b)cisC=a+b+c.
Shivam Reply
give me the waec 2019 questions
Aaron Reply
the polar co-ordinate of the point (-1, -1)
Sumit Reply

### Read also:

#### Get the best Algebra and trigonometry course in your pocket!

Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Algebra and trigonometry' conversation and receive update notifications?

 By By By By By