To simplify the exponential expression [tex]\(27^{\frac{1}{3}}\)[/tex], you are essentially looking for the cube root of 27. Let's break down the steps:
1. Understanding the Power: The expression [tex]\(27^{\frac{1}{3}}\)[/tex] means you need to find a number that, when raised to the power of 3, equals 27.
2. Identifying the Cube Root: Recall that 27 can be written as [tex]\(3^3\)[/tex] because:
[tex]\[ 3 \times 3 \times 3 = 27. \][/tex]
3. Simplification: Since 27 is [tex]\(3^3\)[/tex], taking the cube root of 27 is the same as finding the number:
[tex]\[ (3^3)^{\frac{1}{3}}. \][/tex]
4. Applying Exponent Rules: When you take the cube root of [tex]\(3^3\)[/tex], the exponent rules state that:
[tex]\[ (a^m)^{\frac{1}{m}} = a. \][/tex]
Therefore,
[tex]\[ (3^3)^{\frac{1}{3}} = 3^{3 \times \frac{1}{3}} = 3^1 = 3. \][/tex]
Therefore, the simplified value of the exponential expression [tex]\(27^{\frac{1}{3}}\)[/tex] is [tex]\(3\)[/tex].
Hence, out of the given choices:
- [tex]\(\frac{1}{3}\)[/tex]
- [tex]\(\frac{1}{9}\)[/tex]
- 3
- 9
The correct answer is [tex]\( \boxed{3} \)[/tex].
