To solve the expression [tex]\(\left(5^{\frac{1}{8}} \cdot 5^{\frac{3}{8}}\right)^3\)[/tex], let's break it down step-by-step:
1. Simplify the base expression:
First, we use the property of exponents which states that when multiplying two expressions with the same base, you add their 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]
2. Apply the outer exponent:
Next, we need to raise the simplified base expression to the power of 3. This uses the property of exponents that states [tex]\((a^m)^n = a^{mn}\)[/tex]:
[tex]\[
(5^{\frac{1}{2}})^3 = 5^{\left(\frac{1}{2} \cdot 3\right)} = 5^{\frac{3}{2}}
\][/tex]
Thus, the original expression [tex]\(\left(5^{\frac{1}{8}} \cdot 5^{\frac{3}{8}}\right)^3\)[/tex] simplifies to [tex]\(5^{\frac{3}{2}}\)[/tex].
3. Comparing with given options:
- [tex]\(\mathbf{5^{\frac{3}{2}}}\)[/tex]: This is equivalent to our simplified result, so it is correct.
- [tex]\(\mathbf{5^{\frac{9}{8}}}\)[/tex]: This is not equivalent to [tex]\(5^{\frac{3}{2}}\)[/tex].
- [tex]\(\mathbf{\sqrt{5^3}}\)[/tex]: This expression can be rewritten using exponents as [tex]\(5^{\frac{3}{2}}\)[/tex]. Therefore, this is equivalent to our simplified result.
- [tex]\(\mathbf{(\sqrt[e]{5})^9}\)[/tex]: This expression means raising the e-th root of 5 to the power of 9. Since the base e is not specified and does not match our simplified form, this is not equivalent to [tex]\(5^{\frac{3}{2}}\)[/tex].
Given the detailed steps, the expressions equivalent to [tex]\(\left(5^{\frac{1}{8}} \cdot 5^{\frac{3}{8}}\right)^3\)[/tex] are:
- [tex]\(5^{\frac{3}{2}}\)[/tex]
- [tex]\(\sqrt{5^3}\)[/tex]
These are the two correct equivalent expressions.
