# 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

How look for the general solution of a trig function
stock therom F=(x2+y2) i-2xy J jaha x=a y=o y=b
root under 3-root under 2 by 5 y square
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
cosA\1+sinA=secA-tanA
why two x + seven is equal to nineteen.
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.
simplify each radical by removing as many factors as possible (a) √75
how is infinity bidder from undefined?
what is the value of x in 4x-2+3
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?
v=lbh calculate the volume if i.l=5cm, b=2cm ,h=3cm
Need help with math
Peya
can you help me on this topic of Geometry if l help you
litshani
( cosec Q _ cot Q ) whole spuare = 1_cosQ / 1+cosQ
A guy wire for a suspension bridge runs from the ground diagonally to the top of the closest pylon to make a triangle. We can use the Pythagorean Theorem to find the length of guy wire needed. The square of the distance between the wire on the ground and the pylon on the ground is 90,000 feet. The square of the height of the pylon is 160,000 feet. So, the length of the guy wire can be found by evaluating √(90000+160000). What is the length of the guy wire?
the indicated sum of a sequence is known as