TUTORIALS:
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Rational Exponents
Definition —
Rational Exponent With Numerator m
If a is a real number, m is a nonnegative integer, n is a positive
integer, and
is in lowest terms, then
![](./articles_imgs/927/ration59.gif)
If n is even, then am/n is a real number only when a
≥ 0.
Examples:
![](./articles_imgs/927/ration60.gif)
Note:
When m = 1, we have
![](./articles_imgs/927/ration61.gif)
That is
![](./articles_imgs/927/ration62.gif)
Example 1
Rewrite using a radical and evaluate: 323/5
Solution
We will use three different methods to solve this problem.
Method 1 Use
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323/5 |
Here, a = 32, m = 3, and n = 5. |
![](./articles_imgs/927/ration64.gif) |
Find the prime factorization of 32.
32 = 2 · 2
· 2 · 2
· 2 = 25. |
![](./articles_imgs/927/ration65.gif) |
Simplify the radical.
Evaluate 23. |
= (2)3 = 8 |
Method 2 Use
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Comparing am/n to 323/5 we see that a
= 32,
m = 3, and n = 5. |
![](./articles_imgs/927/ration67.gif) |
Use your calculator to find 323. |
![](./articles_imgs/927/ration68.gif) |
Use your calculator to find
![](./articles_imgs/927/ration69.gif) |
= 8 |
Method 3 Use the Power of a Power Property
of Exponents, (am)n = amn.
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323/5 |
Find the prime factorization of 32.
32 = 2 · 2
· 2 · 2
· 2 = 25. |
= (25)3/5 |
Use the Power of a Power Property of Exponents.
|
![](./articles_imgs/927/ration70.gif) |
Multiply the exponents and write the result
in lowest terms.
Evaluate 23. |
= 23
= 8 |
In summary, all three methods lead to the same result: 323/5
= 8
Note:
Method 2 typically means dealing with
larger numbers since we first evaluate the
power.
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