Let's start with the given expression and simplify it step-by-step. The expression we need to simplify is:
[tex]\[
\left(5^{\frac{1}{8}} \cdot 5^{\frac{3}{8}}\right)^3
\][/tex]
### Step 1: Combine the exponents inside the parentheses
When we multiply two powers of the same base, we add the exponents:
[tex]\[
5^{\frac{1}{8}} \cdot 5^{\frac{3}{8}} = 5^{\left(\frac{1}{8} + \frac{3}{8}\right)} = 5^{\frac{4}{8}} = 5^{\frac{1}{2}}
\][/tex]
### Step 2: Raise the result to the power of 3
Now, we need to raise [tex]\(5^{\frac{1}{2}}\)[/tex] to the power of 3:
[tex]\[
\left(5^{\frac{1}{2}}\right)^3 = 5^{\left(\frac{1}{2} \cdot 3\right)} = 5^{\frac{3}{2}}
\][/tex]
So, the simplified expression is:
[tex]\[
5^{\frac{3}{2}}
\][/tex]
### Step 3: Generate equivalent forms
Now, let's check the equivalence of the given expressions:
#### Expression 1: [tex]\(5^{\frac{3}{2}}\)[/tex]
This matches our simplified result. Thus,
[tex]\[
5^{\frac{3}{2}} \text{ is equivalent to } \left(5^{\frac{1}{8}} \cdot 5^{\frac{3}{8}}\right)^3.
\][/tex]
#### Expression 2: [tex]\(5^{\frac{9}{8}}\)[/tex]
Simplified as a fraction, [tex]\(5^{\frac{9}{8}}\)[/tex] represents a different fractional exponent. It is not equivalent to [tex]\(5^{\frac{3}{2}}\)[/tex].
#### Expression 3: [tex]\(\sqrt{5^3}\)[/tex]
Since the square root of [tex]\(a^b\)[/tex] is [tex]\(a^{\frac{b}{2}}\)[/tex], we get:
[tex]\[
\sqrt{5^3} = 5^{\frac{3}{2}}
\][/tex]
Hence, [tex]\(\sqrt{5^3}\)[/tex] is equivalent to [tex]\(5^{\frac{3}{2}}\)[/tex].
#### Expression 4: [tex]\((\sqrt[8]{5})^9\)[/tex]
Simplifying inside the parentheses, [tex]\(\sqrt[8]{5} = 5^{\frac{1}{8}}\)[/tex], raising this to the 9th power:
[tex]\[
(5^{\frac{1}{8}})^9 = 5^{\frac{1}{8} \cdot 9} = 5^{\frac{9}{8}}
\][/tex]
Hence, [tex]\((\sqrt[8]{5})^9\)[/tex] represents the same expression as [tex]\(5^{\frac{9}{8}}\)[/tex], which we observed was not equivalent to [tex]\(5^{\frac{3}{2}}\)[/tex].
### Conclusion:
The expressions equivalent to [tex]\(\left(5^{\frac{1}{8}} \cdot 5^{\frac{3}{8}}\right)^3\)[/tex] are:
[tex]\[
5^{\frac{3}{2}} \text{ and } \sqrt{5^3}
\][/tex]
