Question 352392
For this problem you need to understand:<ul><li>The relationship between radicals and rational/fractional exponents: {{{q^(a/b) = root(b, q^a) = (root(b, q))^a}}}</li><li>Square roots are "second roots". IOW: {{{sqrt(x)}}} is the same as {{{root(2, x)}}}.</li></ul>
So {{{12sqrt(a^10)}}} written with rational exponents would be:
{{{12*(a^10)^(1/2)}}}
Using the rule for exponents, {{{(a^p)^q = a^((p*q))}}}, we can simplify the "a" part of the above:
{{{12*a^((10*(1/2)))}}}
which simplifies to
{{{12a^5}}}<br>
There is only one thing left. All square roots are supposed to be positive or zero, including {{{sqrt(a^10)}}}. And 12 times a positive number or zero, like 12sqrt(a^10), will be positive or zero. However, our answer at this point is not necessarily positive or zero. In fact, if "a" is negative then {{{12a^5}}} will be negative, too. Since "a" could be negative (there's no reason it can't be), we cannot use {{{12a^5}}} for the answer. We must somehow ensure that our answer must be positive or zero. There are a variety of expressions we could use. Probably the preferred one is found as follows:
{{{12a^5}}}
{{{12a^4*a}}}
{{{12a^4*abs(a)}}}
This expression will always be positive or zero no matter what "a" is.