To evaluate the expression [tex]\(8^{\frac{8}{3}}\)[/tex], we can follow these steps:
1. Express the Base as a Power of a Smaller Prime Number:
First, recognize that 8 can be written as a power of 2:
[tex]\[
8 = 2^3
\][/tex]
2. Rewrite the Original Expression:
Substitute [tex]\(2^3\)[/tex] for 8 in the original expression:
[tex]\[
8^{\frac{8}{3}} = (2^3)^{\frac{8}{3}}
\][/tex]
3. Apply the Power of a Power Rule:
When raising a power to another power, we multiply the exponents:
[tex]\[
(2^3)^{\frac{8}{3}} = 2^{3 \cdot \frac{8}{3}}
\][/tex]
4. Simplify the Exponent:
Simplify the multiplication in the exponent:
[tex]\[
3 \cdot \frac{8}{3} = 8
\][/tex]
Therefore:
[tex]\[
2^{3 \cdot \frac{8}{3}} = 2^8
\][/tex]
5. Evaluate [tex]\(2^8\)[/tex]:
Calculate [tex]\(2^8\)[/tex] (since [tex]\(2^8 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2\)[/tex]):
[tex]\[
2^8 = 256
\][/tex]
6. Final Result:
Thus, the value of [tex]\(8^{\frac{8}{3}}\)[/tex] is:
[tex]\[
8^{\frac{8}{3}} = 256
\][/tex]
Hence, the evaluated result is:
[tex]\[
8^{\frac{8}{3}} = 255.99999999999991
\][/tex]
