C. Rational Law of Exponents, Part 2
Key Lesson Marker
The second form of the Rational Law of Exponents, given the exponent of the power in the form
, is:
The first form of Rational Exponent Law,
, can be used to quickly write a power with a fractional exponent in radical form or vice versa.
To write a power with a rational exponent as a radical, take the denominator of the exponent and place it in the index position of the radical. The numerator becomes the exponent of the power in the radicand.
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Example 3 |
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Write the following expressions in radical form and exponential form, respectively
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You can now apply the Rational Exponent Law to help simplify and evaluate simple exponential or radical expressions without the use of a calculator.
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Example 4 |
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Evaluate the following expressions.
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Using your knowledge of all the Exponent Laws, you can now simplify and evaluate expressions using a combination of Exponent Laws and radicals.
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Example 5 |
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Simplify the following expressions.
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Example 6 |
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Evaluate
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without using a calculator.
In the previous examples, you may have noticed two of the powers had negative bases and rational exponents. Keep in mind that some exponential expressions with negative bases and rational exponents are not defined in the Real Number system. The reason
becomes apparent when the exponential expression is written in radical form. Have you ever tried to take the square root of a negative number on a calculator? If not, try evaluating
in your calculator. Now try evaluating the cube root of a negative number such as
.
Taking the square root of a negative number results in an error message, but taking the cube root of a negative number results in a negative answer. Generally, in order to arrive at a Real Number solution, a radical with an even index cannot have a negative
radicand. A radical with an odd index is not restricted with respect to the value of the radicand.