To solve the expression [tex]\( 8^{\frac{8}{3}} \)[/tex], follow these steps:
1. Identify the base and the exponent: Here, the base is [tex]\( 8 \)[/tex] and the exponent is [tex]\( \frac{8}{3} \)[/tex].
2. Express the exponent in fractional form: The exponent [tex]\( \frac{8}{3} \)[/tex] is already in fractional form. This can be interpreted as raising the base to a power and then taking the cube root.
3. Simplify the base if possible: In this case, the base [tex]\( 8 \)[/tex] can be expressed as [tex]\( 2^3 \)[/tex]. So,
[tex]\[
8 = 2^3
\][/tex]
This transforms the equation to:
[tex]\[
8^{\frac{8}{3}} = (2^3)^{\frac{8}{3}}
\][/tex]
4. Apply the power of a power property: According to the properties of exponents [tex]\((a^m)^n = a^{m \cdot n}\)[/tex]:
[tex]\[
(2^3)^{\frac{8}{3}} = 2^{3 \cdot \frac{8}{3}}
\][/tex]
5. Simplify the exponent: Multiply the exponents:
[tex]\[
3 \cdot \frac{8}{3} = 8
\][/tex]
So, we get:
[tex]\[
2^8
\][/tex]
6. Calculate the final result: Raise [tex]\( 2 \)[/tex] to the power of 8:
[tex]\[
2^8 = 256
\][/tex]
Therefore, the value of [tex]\( 8^{\frac{8}{3}} \)[/tex] is [tex]\( 256 \)[/tex]. Given in the context, due to rounding and precision, the final answer is:
[tex]\[
8^{\frac{8}{3}} \approx 255.99999999999991
\][/tex]
