Let's solve the problem step-by-step to determine which expressions are equivalent to \(\left(5^{\frac{1}{8}} \cdot 5^{\frac{3}{8}}\right)^3\).
### Step 1: Simplify the Inside of the Parentheses
First, we need to simplify the expression inside the parentheses \(\left(5^{\frac{1}{8}} \cdot 5^{\frac{3}{8}}\right)\).
We use the property of exponents: when multiplying like bases, we add the exponents. Therefore,
[tex]\[
5^{\frac{1}{8}} \cdot 5^{\frac{3}{8}} = 5^{\frac{1}{8} + \frac{3}{8}} = 5^{\frac{4}{8}} = 5^{\frac{1}{2}}
\][/tex]
### Step 2: Raise the Simplified Expression to the Power of 3
Next, we raise the simplified expression to the power of 3:
[tex]\[
(5^{\frac{1}{2}})^3
\][/tex]
Using the property of exponents \((a^m)^n = a^{m \cdot n}\):
[tex]\[
(5^{\frac{1}{2}})^3 = 5^{\frac{1}{2} \cdot 3} = 5^{\frac{3}{2}}
\][/tex]
### Step 3: Compare with the Given Expressions
Now, let's compare this result \(5^{\frac{3}{2}}\) with the given expressions one by one to see which are equivalent.
1. \(5^{\frac{3}{2}}\):
- This is exactly what we derived, so it is equivalent.
2. \(5^{\frac{9}{8}}\):
- This is different from \(5^{\frac{3}{2}}\).
3. \(\sqrt{5^3}\):
- This expression can be simplified. Recall that the square root of a number is the same as raising it to the power of \( \frac{1}{2} \):
[tex]\[
\sqrt{5^3} = (5^3)^{\frac{1}{2}} = 5^{3 \cdot \frac{1}{2}} = 5^{\frac{3}{2}}
\][/tex]
- So, \(\sqrt{5^3}\) is equivalent to \(5^{\frac{3}{2}}\).
4. \((\sqrt[8]{5})^9\):
- This expression can be simplified. Recall that the 8th root of a number is the same as raising it to the power of \(\frac{1}{8}\):
[tex]\[
(\sqrt[8]{5})^9 = (5^{\frac{1}{8}})^9 = 5^{\frac{1}{8} \cdot 9} = 5^{\frac{9}{8}}
\][/tex]
- This is different from \(5^{\frac{3}{2}}\).
### Conclusion
The expressions equivalent to \(\left(5^{\frac{1}{8}} \cdot 5^{ \frac{3}{8}}\right)^3\) are:
[tex]\[
5^{\frac{3}{2}} \quad \text{and} \quad \sqrt{5^3}
\][/tex]
