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7.2
Rational Numbers as Exponents
■ Rational Exponents
■ Negative Rational Exponents
■ Laws of Exponents
■ Simplifying Radical Expressions
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Rational Exponents
Consider a1/2a1/2. If we still want to add exponents when multiplying, it must follow that a1/2a1/2 = a1/2 + 1/2, or a1. This suggests that a1/2 is a square root of a.
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When a is nonnegative, n can be any natural number greater than 1. When a is negative, n must be odd.
Note that the denominator of the exponent becomes the index and the base becomes the radicand.
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Example
Write an equivalent expression using radical notation.
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Solution
The denominator of the exponent becomes the index. The base becomes the radicand.
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Write an equivalent expression using exponential notation.
Example
Solution
The index becomes the denominator of the exponent. The radicand becomes the base.
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Positive Rational Exponents
For any natural numbers m and n (n ≠1) and any real number a for which exists,
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Example
Write an equivalent expression using radical notation and simplify.
Solution
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Write an equivalent expression using exponential notation.
Example
Solution
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Negative Rational Exponents
For any rational number m/n and any nonzero real number a for which exists,
Caution! A negative exponent does not indicate that the expression in which it appears is negative.
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Example
Write an equivalent expression with positive exponents and simplify, if possible.
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Solution
8-2/3 is the reciprocal of 82/3.
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Laws of Exponents
The same laws hold for rational exponents as for integer exponents.
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Laws of Exponents
For any real numbers a and b and any rational exponents m and n for which am, an, and bm are defined: 1. 2. 3. 4.
In multiplying, add exponents if the bases are the same.
In dividing, subtract exponents if the bases are the same. (Assume
To raise a power to a power, multiply the exponents.
To raise a product to a power, raise each factor to the power and multiply.
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Example
Use the laws of exponents to simplify.
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Solution
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Simplifying Radical Expressions
To Simplify Radical Expressions
1. Convert radical expressions to exponential
expressions.
2. Use arithmetic and the laws of exponents to simplify.
3. Convert back to radical notation as needed.
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Example
Use rational exponents to simplify. Do not use exponents that are fractions in the final answer.
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Solution
Convert to exponential notation
Simplify the exponent and return to radical notation
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