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 Dependent Variable

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# Quotient Rule for Exponents

We can use arithmetic to simplify the quotient of two exponential expressions. For example,

There are five 2â€™s in the numerator and three 2â€™s in the denominator. After dividing, two 2â€™s remain. The exponent in 22 can be obtained by subtracting the exponents 3 and 5. This example illustrates the quotient rule for exponents.

Quotient Rule for Exponents

If m and n are any integers and a 0, then

Example 1

Using the quotient rule

Simplify each expression. Write answers with positive exponents only. All variables represent nonzero real numbers.

Solution

The next example further illustrates the rules of exponents. Remember that the bases must be identical for the quotient rule or the product rule.

Example 2

Using the product and quotient rules

Use the rules of exponents to simplify each expression. Write answers with positive exponents only. All variables represent nonzero real numbers.

Solution

 = 2x0 Quotient rule: -7 - (-7) = 0 = 2 Definition of zero exponent
 Product rule: w1 Â· w-4 = w-3 Quotient rule: -3 - (-2) = -1 Definition of negative exponent