Question

What is the following product? 3 root 5 times root 2

Answer 1

Answer:

$\sqrt[6]{200}$

Step-by-step explanation:

First of all, you need to know that you cannot multiply those radicals without having a common index (the little number outside the radical that sits in the curve of the radical).  One is a 3 and the other, without being stated outright, is understood to be a 2.  BUT we can make them like.  The index of a radical is the denominator of the exponential equivalent.  

$\sqrt[3]{5}=5^{\frac{1}{3}}$ and

$\sqrt{2}=2^{\frac{1}{2}}$

See how the indexes are now the denominators of the rational exponents.  We can make them like by finding the LCM of 3 and 2...which is 6:

$5^{\frac{1}{3}}=5^{\frac{2}{6}}$ and

$2^{\frac{1}{2}}= 2^{\frac{3}{6}}$

Now that the indexes are like, we rewrite them as radicals again:

$\sqrt[6]{5^2}*\sqrt[6]{2^3}$ which, simplified, is

$\sqrt[6]{25}*\sqrt[6]{8}$

Now we can find the product which is

$\sqrt[6]{200}$

Answer 2

$\bf \textit{rational exponents} \\\\ a^{\frac{ n}{ m}} \implies \sqrt[ m]{a^ n} ~\hspace{10em} \\\\[-0.35em] \rule{34em}{0.25pt}\\\\ \sqrt[3]{5}\cdot \sqrt{2}\implies \sqrt[3]{5^1}\cdot \sqrt[2]{2^1}\implies 5^{\frac{1}{3}}\cdot 2^{\frac{1}{2}}\implies 5^{\frac{2}{6}}\cdot 2^{\frac{3}{6}}\implies \sqrt[6]{5^2}\cdot \sqrt[6]{2^3} \\\\\\ \sqrt[6]{25}\cdot \sqrt[6]{8}\implies \sqrt[6]{25\cdot 8}\implies \sqrt[6]{200}$