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Write each of the following products with a single base. Do not simplify further.

  1. ( ( 3 y ) 8 ) 3
  2. ( t 5 ) 7
  3. ( ( g ) 4 ) 4
  1. ( 3 y ) 24
  2. t 35
  3. ( g ) 16
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Using the zero exponent rule of exponents

Return to the quotient rule. We made the condition that m > n so that the difference m n would never be zero or negative. What would happen if m = n ? In this case, we would use the zero exponent rule of exponents to simplify the expression to 1. To see how this is done, let us begin with an example.

t 8 t 8 = t 8 t 8 = 1

If we were to simplify the original expression using the quotient rule, we would have

t 8 t 8 = t 8 8 = t 0

If we equate the two answers, the result is t 0 = 1. This is true for any nonzero real number, or any variable representing a real number.

a 0 = 1

The sole exception is the expression 0 0 . This appears later in more advanced courses, but for now, we will consider the value to be undefined.

The zero exponent rule of exponents

For any nonzero real number a , the zero exponent rule of exponents states that

a 0 = 1

Using the zero exponent rule

Simplify each expression using the zero exponent rule of exponents.

  1. c 3 c 3
  2. −3 x 5 x 5
  3. ( j 2 k ) 4 ( j 2 k ) ( j 2 k ) 3
  4. 5 ( r s 2 ) 2 ( r s 2 ) 2

Use the zero exponent and other rules to simplify each expression.


  1. c 3 c 3 = c 3 3 = c 0 = 1

  2. −3 x 5 x 5 = −3 x 5 x 5 = −3 x 5 5 = −3 x 0 = −3 1 = −3

  3. ( j 2 k ) 4 ( j 2 k ) ( j 2 k ) 3 = ( j 2 k ) 4 ( j 2 k ) 1 + 3 Use the product rule in the denominator . = ( j 2 k ) 4 ( j 2 k ) 4 Simplify . = ( j 2 k ) 4 4 Use the quotient rule . = ( j 2 k ) 0 Simplify . = 1

  4. 5 ( r s 2 ) 2 ( r s 2 ) 2 = 5 ( r s 2 ) 2 2 Use the quotient rule . = 5 ( r s 2 ) 0 Simplify . = 5 1 Use the zero exponent rule . = 5 Simplify .
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Simplify each expression using the zero exponent rule of exponents.

  1. t 7 t 7
  2. ( d e 2 ) 11 2 ( d e 2 ) 11
  3. w 4 w 2 w 6
  4. t 3 t 4 t 2 t 5
  1. 1
  2. 1 2
  3. 1
  4. 1
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Using the negative rule of exponents

Another useful result occurs if we relax the condition that m > n in the quotient rule even further. For example, can we simplify h 3 h 5 ? When m < n —that is, where the difference m n is negative—we can use the negative rule of exponents to simplify the expression to its reciprocal.

Divide one exponential expression by another with a larger exponent. Use our example, h 3 h 5 .

h 3 h 5 = h h h h h h h h = h h h h h h h h = 1 h h = 1 h 2

If we were to simplify the original expression using the quotient rule, we would have

h 3 h 5 = h 3 5 =   h −2

Putting the answers together, we have h −2 = 1 h 2 . This is true for any nonzero real number, or any variable representing a nonzero real number.

A factor with a negative exponent becomes the same factor with a positive exponent if it is moved across the fraction bar—from numerator to denominator or vice versa.

a n = 1 a n and a n = 1 a n

We have shown that the exponential expression a n is defined when n is a natural number, 0, or the negative of a natural number. That means that a n is defined for any integer n . Also, the product and quotient rules and all of the rules we will look at soon hold for any integer n .

The negative rule of exponents

For any nonzero real number a and natural number n , the negative rule of exponents states that

a n = 1 a n

Using the negative exponent rule

Write each of the following quotients with a single base. Do not simplify further. Write answers with positive exponents.

  1. θ 3 θ 10
  2. z 2 z z 4
  3. ( −5 t 3 ) 4 ( −5 t 3 ) 8
  1. θ 3 θ 10 = θ 3 10 = θ −7 = 1 θ 7
  2. z 2 z z 4 = z 2 + 1 z 4 = z 3 z 4 = z 3 4 = z −1 = 1 z
  3. ( −5 t 3 ) 4 ( −5 t 3 ) 8 = ( −5 t 3 ) 4 8 = ( −5 t 3 ) −4 = 1 ( −5 t 3 ) 4
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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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