To find [tex]\(27^{\frac{4}{3}}\)[/tex], we can follow these steps:
1. First, recall that the exponent [tex]\(\frac{4}{3}\)[/tex] can be broken down into two parts: the cube root (i.e., raising to the power of [tex]\(\frac{1}{3}\)[/tex]) and then raising to the power of 4.
2. We know that [tex]\(27\)[/tex] can be expressed as [tex]\(3^3\)[/tex]. Therefore, we can write:
[tex]\[
27 = 3^3
\][/tex]
3. Applying the exponent [tex]\(\frac{4}{3}\)[/tex] to [tex]\(27\)[/tex]:
[tex]\[
27^{\frac{4}{3}} = (3^3)^{\frac{4}{3}}
\][/tex]
4. Using the property of exponents [tex]\((a^m)^n = a^{m \cdot n}\)[/tex], we can simplify:
[tex]\[
(3^3)^{\frac{4}{3}} = 3^{3 \cdot \frac{4}{3}} = 3^4
\][/tex]
5. Calculate [tex]\(3^4\)[/tex]:
[tex]\[
3^4 = 3 \times 3 \times 3 \times 3 = 81
\][/tex]
Hence, the value of [tex]\(27^{\frac{4}{3}}\)[/tex] is [tex]\(81\)[/tex].
The options given are:
- 12
- 108
- 81
- 4
The correct answer is:
[tex]\[
\boxed{81}
\][/tex]
