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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

if tan alpha + beta is equal to sin x + Y then prove that X square + Y square - 2 I got hyperbole 2 Beta + 1 is equal to zero
Rahul Reply
sin^4+sin^2=1, prove that tan^2-tan^4+1=0
SAYANTANI Reply
what is the formula used for this question? "Jamal wants to save $54,000 for a down payment on a home. How much will he need to invest in an account with 8.2% APR, compounding daily, in order to reach his goal in 5 years?"
Kuz Reply
i don't need help solving it I just need a memory jogger please.
Kuz
A = P(1 + r/n) ^rt
Dale
how to solve an expression when equal to zero
Mintah Reply
its a very simple
Kavita
gave your expression then i solve
Kavita
Hy guys, I have a problem when it comes on solving equations and expressions, can you help me 😭😭
Thuli
Tomorrow its an revision on factorising and Simplifying...
Thuli
ok sent the quiz
kurash
send
Kavita
Hi
Masum
What is the value of log-1
Masum
the value of log1=0
Kavita
Log(-1)
Masum
What is the value of i^i
Masum
log -1 is 1.36
kurash
No
Masum
no I m right
Kavita
No sister.
Masum
no I m right
Kavita
tan20°×tan30°×tan45°×tan50°×tan60°×tan70°
Joju Reply
jaldi batao
Joju
Find the value of x between 0degree and 360 degree which satisfy the equation 3sinx =tanx
musah Reply
what is sine?
tae Reply
what is the standard form of 1
Sanjana Reply
1×10^0
Akugry
Evalute exponential functions
Sujata Reply
30
Shani
The sides of a triangle are three consecutive natural number numbers and it's largest angle is twice the smallest one. determine the sides of a triangle
Jaya Reply
Will be with you shortly
Inkoom
3, 4, 5 principle from geo? sounds like a 90 and 2 45's to me that my answer
Neese
answer is 2, 3, 4
Gaurav
prove that [a+b, b+c, c+a]= 2[a b c]
Ashutosh Reply
can't prove
Akugry
i can prove [a+b+b+c+c+a]=2[a+b+c]
this is simple
Akugry
hi
Stormzy
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
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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