which expression(s) are equivalent to $(5^{\frac{1}{8}}cdot5^{\frac{3}{8}})^3$?\n$5^{\frac{3}{2}}$\n$5^{\frac…

Explanation:

Step1: Use exponent - product rule

According to the rule (a^m\cdot a^n=a^{m + n}), for (5^{\frac{1}{8}}\cdot5^{\frac{3}{8}}), we have (5^{\frac{1}{8}+\frac{3}{8}}=5^{\frac{4}{8}} = 5^{\frac{1}{2}}).

Step2: Apply power - of - a - power rule

Now we have ((5^{\frac{1}{2}})^3). According to the rule ((a^m)^n=a^{mn}), then ((5^{\frac{1}{2}})^3=5^{\frac{1}{2}\times3}=5^{\frac{3}{2}}).

Step3: Rewrite in radical form

Since (a^{\frac{m}{n}}=\sqrt[n]{a^m}), for (5^{\frac{3}{2}}), it is (\sqrt{5^3}). Also, (5^{\frac{3}{2}}=(5^{\frac{1}{8}})^9 = (\sqrt[8]{5})^9).

Answer:

A. (5^{\frac{3}{2}}), C. (\sqrt{5^3}), D. ((\sqrt[8]{5})^9)