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 Dependent Variable

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# Properties of Rational Exponents

The properties of integer exponents also hold for rational exponents. This table contains an example of each property.

 Property Integer Exponents Rational Exponents Multiplication 32 Â· 34 = 32 + 4 = 36 37/5 Â· 33/5 = 37/5 + 3/5 = 310/5 = 32 = 9 Division Power of a Power (52)3 = 52 Â· 3 = 56 Power of a Product (5 Â· 7)3 = 53 Â· 73 (5 Â· 7)1/6 = 51/6 Â· 71/6 Power of a Quotient

The Power of a Power Property is particularly useful when we work with rational exponents.

 For example, letâ€™s use this property to rewrite 53/4 in two different, but equivalent, ways. Notation 153/4 Notation 253/4 Rewrite the fraction 3/4 as 3 Â· (1/4) and as (1/4) Â· 3. = 53 Â· (1/4) = 5(1/4) Â· 3 Use the Power of a Power Property. = (53)1/4 = (51/4)3 Use radical notation.

Thus,

Letâ€™s compare the original exponential expression, 53/4, with the final radical expressions,

â€¢ In the original expression 53/4, the exponent is .

â€¢ The denominator, 4, is the index of the radical .

â€¢ The numerator, 3, is a power in the radical expression.

This relationship is true in general.