C. Rational Law of Exponents
Key Lesson Marker
The Rational Law of Exponents, given the exponent of the power in the form
, is:
One of the applications of the Rational Law of Exponents is to express powers with rational exponents as radicals and vice versa.
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Example 1 |
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Write the following expressions in exponential form and radical form respectively.
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The Rational Law of Exponents is also helpful when simplifying and evaluating powers with rational exponents.
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Example 2 |
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Evaluate the following powers.
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What rule can be defined when a power has a rational exponent whose numerator is not 1? For example,
.
By applying the Power of Power Law in reverse, the expression can be rewritten as "a cubed to the power of
".
Since taking a value to the exponent of
is the same as taking the square root of the value,
can also be written as a radical!
In this case, you are taking the square root of a cubed.
Notice that
could also have been written as
a to the power of
, cubed:
.
Applying the Rational Exponent Law to
, results in
. In this scenario, you are cubing the square root of
a.
The order in which the Power of Power Law is applied does not matter because it results in equivalent expressions. Therefore,
.
To summarize, if given a power of the form
, it is equivalent to taking
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a to the mth power and then taking the nth root:
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the nth root of a, then taking the mth power:
