# 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

f(x)=x/x+2 given g(x)=1+2x/1-x show that gf(x)=1+2x/3
proof
AUSTINE
sebd me some questions about anything ill solve for yall
how to solve x²=2x+8 factorization?
x=2x+8 x-2x=2x+8-2x x-2x=8 -x=8 -x/-1=8/-1 x=-8 prove: if x=-8 -8=2(-8)+8 -8=-16+8 -8=-8 (PROVEN)
Manifoldee
x=2x+8
Manifoldee
×=2x-8 minus both sides by 2x
Manifoldee
so, x-2x=2x+8-2x
Manifoldee
then cancel out 2x and -2x, cuz 2x-2x is obviously zero
Manifoldee
so it would be like this: x-2x=8
Manifoldee
then we all know that beside the variable is a number (1): (1)x-2x=8
Manifoldee
so we will going to minus that 1-2=-1
Manifoldee
so it would be -x=8
Manifoldee
so next step is to cancel out negative number beside x so we get positive x
Manifoldee
so by doing it you need to divide both side by -1 so it would be like this: (-1x/-1)=(8/-1)
Manifoldee
so -1/-1=1
Manifoldee
so x=-8
Manifoldee
Manifoldee
so we should prove it
Manifoldee
x=2x+8 x-2x=8 -x=8 x=-8 by mantu from India
mantu
lol i just saw its x²
Manifoldee
x²=2x-8 x²-2x=8 -x²=8 x²=-8 square root(x²)=square root(-8) x=sq. root(-8)
Manifoldee
I mean x²=2x+8 by factorization method
Kristof
I think x=-2 or x=4
Kristof
x= 2x+8 ×=8-2x - 2x + x = 8 - x = 8 both sides divided - 1 -×/-1 = 8/-1 × = - 8 //// from somalia
Mohamed
hii
Amit
how are you
Dorbor
well
Biswajit
can u tell me concepts
Gaurav
Find the possible value of 8.5 using moivre's theorem
which of these functions is not uniformly cintinuous on (0, 1)? sinx
which of these functions is not uniformly continuous on 0,1
solve this equation by completing the square 3x-4x-7=0
X=7
Muustapha
=7
mantu
x=7
mantu
3x-4x-7=0 -x=7 x=-7
Kr
x=-7
mantu
9x-16x-49=0 -7x=49 -x=7 x=7
mantu
what's the formula
Modress
-x=7
Modress
new member
siame
what is trigonometry
deals with circles, angles, and triangles. Usually in the form of Soh cah toa or sine, cosine, and tangent
Thomas
solve for me this equational y=2-x
what are you solving for
Alex
solve x
Rubben
you would move everything to the other side leaving x by itself. subtract 2 and divide -1.
Nikki
then I got x=-2
Rubben
it will b -y+2=x
Alex
goodness. I'm sorry. I will let Alex take the wheel.
Nikki
ouky thanks braa
Rubben
I think he drive me safe
Rubben
how to get 8 trigonometric function of tanA=0.5, given SinA=5/13? Can you help me?m
More example of algebra and trigo
What is Indices
If one side only of a triangle is given is it possible to solve for the unkown two sides?
cool
Rubben
kya
Khushnama
please I need help in maths
Okey tell me, what's your problem is?
Navin