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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

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cosx.cos2x.cos4x.cos8x
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sinx sin2x is linearly dependent
cr Reply
what is a reciprocal
Ajibola Reply
The reciprocal of a number is 1 divided by a number. eg the reciprocal of 10 is 1/10 which is 0.1
Shemmy
 Reciprocal is a pair of numbers that, when multiplied together, equal to 1. Example; the reciprocal of 3 is ⅓, because 3 multiplied by ⅓ is equal to 1
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each term in a sequence below is five times the previous term what is the eighth term in the sequence
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How look for the general solution of a trig function
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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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