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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

sin theta=3/4.prove that sec square theta barabar 1 + tan square theta by cosec square theta minus cos square theta
Umesh Reply
I want to know trigonometry but I can't understand it anyone who can help
Siyabonga Reply
Yh
Idowu
which part of trig?
Nyemba
functions
Siyabonga
trigonometry
Ganapathi
differentiation doubhts
Ganapathi
hi
Ganapathi
hello
Brittany
Prove that 4sin50-3tan 50=1
Sudip Reply
f(x)= 1 x    f(x)=1x  is shifted down 4 units and to the right 3 units.
Sebit Reply
f (x) = −3x + 5 and g (x) = x − 5 /−3
Sebit
what are real numbers
Marty Reply
I want to know partial fraction Decomposition.
Adama Reply
classes of function in mathematics
Yazidu Reply
divide y2_8y2+5y2/y2
Sumanth Reply
wish i knew calculus to understand what's going on 🙂
Dashawn Reply
@dashawn ... in simple terms, a derivative is the tangent line of the function. which gives the rate of change at that instant. to calculate. given f(x)==ax^n. then f'(x)=n*ax^n-1 . hope that help.
Christopher
thanks bro
Dashawn
maybe when i start calculus in a few months i won't be that lost 😎
Dashawn
what's the derivative of 4x^6
Axmed Reply
24x^5
James
10x
Axmed
24X^5
Taieb
Thanks for this helpfull app
Axmed Reply
secA+tanA=2√5,sinA=?
richa Reply
tan2a+tan2a=√3
Rahulkumar
classes of function
Yazidu
if sinx°=sin@, then @ is - ?
NAVJIT Reply
the value of tan15°•tan20°•tan70°•tan75° -
NAVJIT
0.037 than find sin and tan?
Jon Reply
cos24/25 then find sin and tan
Deepak 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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