what is the following product?\n$sqrt3{5}cdotsqrt{2}$\n$sqrt6{10}$\n$sqrt6{200}$\n$sqrt6{500}$\n$sqrt6{100000…

Explanation:

Step1: Rewrite radicals as exponents

$\sqrt[3]{5}=5^{\frac{1}{3}}$ and $\sqrt{2}=2^{\frac{1}{2}}$ So, $\sqrt[3]{5}\cdot\sqrt{2}=5^{\frac{1}{3}}\cdot2^{\frac{1}{2}}$

Step2: Find a common - denominator for the exponents

The common denominator of 3 and 2 is 6. $5^{\frac{1}{3}} = 5^{\frac{2}{6}}$ and $2^{\frac{1}{2}}=2^{\frac{3}{6}}$ Then $5^{\frac{2}{6}}\cdot2^{\frac{3}{6}}=(5^{2}\cdot2^{3})^{\frac{1}{6}}$

Step3: Calculate the values inside the parentheses

$5^{2}=25$ and $2^{3}=8$ $5^{2}\cdot2^{3}=25\times8 = 200$ So, $(5^{2}\cdot2^{3})^{\frac{1}{6}}=\sqrt[6]{200}$

Answer:

$\sqrt[6]{200}$