Question

select the expression that is equivalent to \\(\sqrt{3} \cdot \sqrt5{3^4}\\). answer \\(3^{\frac{13}{10}}\\) \\(3^{\frac{5}{2}}\\) \\(3^{\frac{2}{5}}\\) \\(3^{\frac{10}{13}}\\)

Explanation:

Step1: Convert radicals to exponents

Recall that $\sqrt[n]{a^m}=a^{\frac{m}{n}}$ and $\sqrt{a}=a^{\frac{1}{2}}$. So, $\sqrt{3} = 3^{\frac{1}{2}}$ and $\sqrt[5]{3^4}=3^{\frac{4}{5}}$.

Step2: Multiply the exponential expressions

When multiplying expressions with the same base, we add the exponents. So, $3^{\frac{1}{2}}\cdot3^{\frac{4}{5}} = 3^{\frac{1}{2}+\frac{4}{5}}$.

Step3: Add the fractions

Find a common denominator for $\frac{1}{2}$ and $\frac{4}{5}$, which is 10. Then $\frac{1}{2}=\frac{5}{10}$ and $\frac{4}{5}=\frac{8}{10}$. So, $\frac{1}{2}+\frac{4}{5}=\frac{5 + 8}{10}=\frac{13}{10}$. Thus, $3^{\frac{1}{2}+\frac{4}{5}}=3^{\frac{13}{10}}$.

Answer:

A. $3^{\frac{13}{10}}$