Question 392993
I assume you mean the 12th root of 64:
{{{root(12, 64)}}}
When simplifying radicals the first thing you look for are factors of the radicand (the expression inside a radical) that are powers of the type of root. In this case we would look for factors of 64 that are powers of 12. Since factoring out a 1 is rarely useful we start by checking 2. {{{2^12 = 4096}}}. And obviously higher numbers to the 12th power will be even higher. So there are no factors of 64 that are 12th powers of anything (except 1).<br>
Another way to simplify radicals can be done once you've learned about fractional exponents. If the radicand is a power of some sumber, write it that way. Since {{{64 = 8^2 = 4^3 = 2^6}}} we have 3 choices as to how we rewrite 64. In this case choose an exponent that is a factor of the root. But all these exponents, 2, 3 and 6, are all factors of 12. In this case choose the smallest base. So we will use {{{2^6}}}:
{{{root(12, 2^6)}}}
Next we rewrite the radical with a fractional exponent:
{{{2^((8/12))}}}
This fraction reduces. (This is why we look for an exponent that is a factor of the type of root.)
{{{2^((1/2))}}}
Now we can rewrite it back in radical form:
{{{root(2, 2^1)}}}
or
{{{sqrt(2)}}}