1. The laws of exponents can be applied to fractional or rational exponents in the same way as integral exponents. 2. Examples are provided to simplify expressions using the laws of exponents to write fractional exponents with positive exponents only and to evaluate expressions with fractional exponents. 3. The examples show applying the power of a power law, expressing bases in exponential form, and simplifying results.

2. Let’s Learn
The laws of exponents for integral exponents are
applicable also for fractional or rational exponents.
Study the following examples.
Example 1. Simplify using the laws of exponents.
Express the answer using positive exponents only.

3. 1. a
1
3 · a
2
3 =
2.
12a
3
5
14a
1
5
=
a
1
3 · a
2
3 = a
1
3 +
2
3
= a
3
3
= a1
= a
12a
3
5
14a
1
5
=
6
7
a
3
5 -
1
5
=
𝟔
𝟕
a
𝟐
𝟓

4. 3. 4𝑎
3
4 2 = 4. 𝑥
1
6 3 =
4𝑎
3
4 2 = 42 a
3
4 2
= 16a
6
4
= 16a
𝟑
𝟐
𝑥
1
6 3 = x
3
6
= x
𝟏
𝟐

5. Example 2. Evaluate 27
2
3 .
Steps Solution
1. Express the base in
exponential form. (27 = 33)
27
2
3 = (33)
2
3
2. Use the power of a power
law of exponents.
= (33)
2
3 or 3
6
3 =
= 32
= 9
3. Simplify the result by
applying the laws of
exponents.
So, 27
2
3 = 9

6. Example 3. Evaluate (32)
4
5
Steps Solution
1. Express the base in
exponential form. (32 = 25)
(32)
4
5 = (25)
4
5
2. Use the power of a power
law of exponents.
= (25)
4
5 or 2
20
5
= 24
= 16
3. Simplify the result by
applying the laws of
exponents.
So, (32)
4
5 = 16

7. Example 4. Evaluate (81)
5
4
Steps Solution
1. Express the base in
exponential form. (81 = 34)
(81)
5
4 = (34)
5
4
2. Use the power of a power
law of exponents.
= (34)
5
4 or 3
20
4
= 35
= 243
3. Simplify the result by
applying the laws of
exponents.
So, (81)
5
4 = 243

8. Example 5. Evaluate (−32)
3
5 .
Steps Solution
1. Express the base in
exponential form. (-32 = -25)
(−32)
3
5 = (−25)
3
5
2. Use the power of a power
law of exponents.
= (−25)
3
5 or −2
15
5
= -23
= -8
3. Simplify the result by
applying the laws of
exponents.
So, (−32)
3
5 = -8