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Key equations

Rules of Exponents
For nonzero real numbers a and b and integers m and n
Product rule a m a n = a m + n
Quotient rule a m a n = a m n
Power rule ( a m ) n = a m n
Zero exponent rule a 0 = 1
Negative rule a n = 1 a n
Power of a product rule ( a b ) n = a n b n
Power of a quotient rule ( a b ) n = 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] .

Section exercises

Verbal

Is 2 3 the same as 3 2 ? Explain.

No, the two expressions are not the same. An exponent tells how many times you multiply the base. So 2 3 is the same as 2 × 2 × 2 , which is 8. 3 2 is the same as 3 × 3 , which is 9.

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When can you add two exponents?

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What is the purpose of scientific notation?

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

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Explain what a negative exponent does.

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Numeric

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

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

4 2 × 4 3 ÷ 4 −4

4 9

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( 12 3 × 12 ) 10

12 40

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7 −6 × 7 −3

1 7 9

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

0.0000314

3.14 × 10 5

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

1.6 × 10 10

16,000,000,000

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Algebraic

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

a b 2 ÷ d −3

a b 2 d 3

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p −4 q 2 p 2 q −3

q 5 p 6

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( y 7 ) 3 ÷ x 14

y 21 x 14

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( 9 z 3 ) −2 y

y 81 z 6

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Real-world applications

To reach escape velocity, a rocket must travel at the rate of 2.2 × 10 6 ft/min. Rewrite the rate in standard notation.

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A dime is the thinnest coin in U.S. currency. A dime’s thickness measures 1.35 × 10 −3 m. Rewrite the number in standard notation.

0.00135 m

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

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

1.0995 × 10 12

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The Gross Domestic Product (GDP) for the United States in the first quarter of 2014 was $ 1.71496 × 10 13 . Rewrite the GDP in standard notation.

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One picometer is approximately 3.397 × 10 −11 in. Rewrite this length using standard notation.

0.00000000003397 in.

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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.

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Technology

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

( 12 3 m 33 4 −3 ) 2

12,230,590,464 m 66

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Extensions

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

( 3 2 a 3 ) −2 ( a 4 2 2 ) 2

a 14 1296

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( 6 2 −24 ) 2 ÷ ( x y ) −5

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m 2 n 3 a 2 c −3 a −7 n −2 m 2 c 4

n a 9 c

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( x 6 y 3 x 3 y −3 y −7 x −3 ) 10

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( ( a b 2 c ) −3 b −3 ) 2

1 a 6 b 6 c 6

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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 6.0221413 × 10 23 . Write Avogadro’s constant in standard notation.

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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 6.62606957 × 10 −34 . Write Planck’s constant in standard notation.

0.000000000000000000000000000000000662606957

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Questions & Answers

x exposant 4 + 4 x exposant 3 + 8 exposant 2 + 4 x + 1 = 0
HERVE Reply
x exposent4+4x exposent3+8x exposent2+4x+1=0
HERVE
How can I solve for a domain and a codomains in a given function?
Oliver Reply
ranges
EDWIN
Thank you I mean range sir.
Oliver
proof for set theory
Kwesi Reply
don't you know?
Inkoom
find to nearest one decimal place of centimeter the length of an arc of circle of radius length 12.5cm and subtending of centeral angle 1.6rad
Martina Reply
factoring polynomial
Noven Reply
what's your topic about?
Shin Reply
find general solution of the Tanx=-1/root3,secx=2/root3
Nani Reply
find general solution of the following equation
Nani
the value of 2 sin square 60 Cos 60
Sanjay Reply
0.75
Lynne
0.75
Inkoom
when can I use sin, cos tan in a giving question
duru Reply
depending on the question
Nicholas
I am a carpenter and I have to cut and assemble a conventional roof line for a new home. The dimensions are: width 30'6" length 40'6". I want a 6 and 12 pitch. The roof is a full hip construction. Give me the L,W and height of rafters for the hip, hip jacks also the length of common jacks.
John
I want to learn the calculations
Koru Reply
where can I get indices
Kojo Reply
I need matrices
Nasasira
hi
Raihany
Hi
Solomon
need help
Raihany
maybe provide us videos
Nasasira
about complex fraction
Raihany
Hello
Cromwell
a
Amie
What do you mean by a
Cromwell
nothing. I accidentally press it
Amie
you guys know any app with matrices?
Khay
Ok
Cromwell
Solve the x? x=18+(24-3)=72
Leizel Reply
x-39=72 x=111
Suraj
Solve the formula for the indicated variable P=b+4a+2c, for b
Deadra Reply
Need help with this question please
Deadra
b=-4ac-2c+P
Denisse
b=p-4a-2c
Suddhen
b= p - 4a - 2c
Snr
p=2(2a+C)+b
Suraj
b=p-2(2a+c)
Tapiwa
P=4a+b+2C
COLEMAN
b=P-4a-2c
COLEMAN
like Deadra, show me the step by step order of operation to alive for b
John
A laser rangefinder is locked on a comet approaching Earth. The distance g(x), in kilometers, of the comet after x days, for x in the interval 0 to 30 days, is given by g(x)=250,000csc(π30x). Graph g(x) on the interval [0, 35]. Evaluate g(5)  and interpret the information. What is the minimum distance between the comet and Earth? When does this occur? To which constant in the equation does this correspond? Find and discuss the meaning of any vertical asymptotes.
Kaitlyn Reply
The sequence is {1,-1,1-1.....} has
amit Reply
prove that [a+b, b+c, c+a]= 2[a b c]
Ashutosh Reply
Practice Key Terms 1

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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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