Certainly! Let's go through the detailed, step-by-step solution to determine which expressions are equivalent to [tex]\(\left(5^{\frac{1}{8}} \cdot 5^{\frac{3}{8}}\right)^3\)[/tex].
1. Simplify the expression inside the parentheses:
[tex]\[
5^{\frac{1}{8}} \cdot 5^{\frac{3}{8}}
\][/tex]
When multiplying expressions with the same base, we add the exponents:
[tex]\[
5^{\frac{1}{8}} \cdot 5^{\frac{3}{8}} = 5^{\frac{1}{8} + \frac{3}{8}}
\][/tex]
2. Add the exponents:
[tex]\[
\frac{1}{8} + \frac{3}{8} = \frac{4}{8} = \frac{1}{2}
\][/tex]
So, the expression simplifies to:
[tex]\[
5^{\frac{1}{2}}
\][/tex]
3. Now consider the entire expression:
[tex]\[
\left(5^{\frac{1}{2}}\right)^3
\][/tex]
When raising a power to another power, we multiply the exponents:
[tex]\[
\left(5^{\frac{1}{2}}\right)^3 = 5^{\frac{1}{2} \cdot 3} = 5^{\frac{3}{2}}
\][/tex]
Thus, the simplified expression is:
[tex]\[
5^{\frac{3}{2}}
\][/tex]
We can now compare this with the given options:
- Option 1: [tex]\(5^{\frac{3}{2}}\)[/tex] is identical to our simplified result, so it is equivalent.
- Option 2: [tex]\(5^{\frac{9}{8}}\)[/tex] is not equivalent to our simplified result, as [tex]\(\frac{9}{8}\)[/tex] is different from [tex]\(\frac{3}{2}\)[/tex].
Therefore, the expression [tex]\(\left(5^{\frac{1}{8}} \cdot 5^{\frac{3}{8}}\right)^3\)[/tex] is equivalent to [tex]\(5^{\frac{3}{2}}\)[/tex].
