Sure! Let's simplify each of the given expressions step-by-step.
1. Simplify [tex]\(\left(8^{\frac{2}{3}}\right)^{4}\)[/tex]:
We use the power of a power property [tex]\((a^m)^n = a^{m \cdot n}\)[/tex].
[tex]\[
\left(8^{\frac{2}{3}}\right)^{4} = 8^{\frac{2}{3} \cdot 4} = 8^{\frac{8}{3}}
\][/tex]
Therefore, [tex]\(\left(8^{\frac{2}{3}}\right)^4 = 8^{\frac{8}{3}}\approx 255.99999999999991\)[/tex].
2. Simplify [tex]\(8^{\frac{8}{3}}\)[/tex]:
[tex]\[
8^{\frac{8}{3}}\approx 255.99999999999991
\][/tex]
3. Simplify [tex]\(8^{\frac{6}{3}}\)[/tex]:
We simplify the fraction [tex]\(\frac{6}{3}\)[/tex] first:
[tex]\[
8^{\frac{6}{3}} = 8^{2}
\][/tex]
And since [tex]\(8^2 = 64\)[/tex],
[tex]\[
8^{\frac{6}{3}} = 64.0
\][/tex]
4. Simplify [tex]\(8^{\frac{6}{12}}\)[/tex]:
We simplify the fraction [tex]\(\frac{6}{12}\)[/tex]:
[tex]\[
8^{\frac{6}{12}} = 8^{\frac{1}{2}}
\][/tex]
This is the same as the square root of 8:
[tex]\[
8^{\frac{1}{2}} = \sqrt{8}
\][/tex]
And since [tex]\(\sqrt{8} = 2\sqrt{2}\)[/tex]:
[tex]\[
2\sqrt{2} \approx 2.8284271247461903
\][/tex]
So, the simplified values are:
[tex]\[
\left(8^{\frac{2}{3}}\right)^{4} \approx 255.99999999999991, \quad 8^{\frac{8}{3}} \approx 255.99999999999991, \quad 8^{\frac{6}{3}} = 64.0, \quad 8^{\frac{6}{12}} \approx 2.8284271247461903
\][/tex]
