which expressions are equivalent to (left(5^{\frac{1}{4}} cdot 5^{\frac{3}{8}}\right)^{\frac{3}{2}})? choose…

Explanation:

Step1: Add exponents inside parentheses

When multiplying exponents with the same base, add the exponents: $5^{\frac{1}{8}} \cdot 5^{\frac{3}{8}} = 5^{\frac{1}{8}+\frac{3}{8}} = 5^{\frac{4}{8}} = 5^{\frac{1}{2}}$

Step2: Multiply by outer exponent

When raising a power to a power, multiply the exponents: $\left(5^{\frac{1}{2}}\right)^3 = 5^{\frac{1}{2} \times 3} = 5^{\frac{3}{2}}$

Step3: Rewrite as radical form

A fractional exponent $\frac{a}{b}$ equals $\sqrt[b]{5^a}$, so: $5^{\frac{3}{2}} = \sqrt{5^3}$

Step4: Verify other equivalent form

First, re-express the original result $5^{\frac{3}{2}}$ with denominator 8: $5^{\frac{3}{2}} = 5^{\frac{12}{8}}$, and $\left(\sqrt[8]{5}\right)^9 = 5^{\frac{9}{8}}$ is not equivalent. $5^{\frac{9}{8}}$ is not equal to $5^{\frac{3}{2}}$.

Answer:

1. $5^{\frac{3}{2}}$
2. $\sqrt{5^3}$