Question 404469
{{{root(4, 6a^12*b^5) * sqrt(6a^12*b^5)}}}
When multiplying radicals you usually use the property of radicals: {{{root(a, p) * root(a, q) = root(a, p*q)}}}. But this property requires that the two radicals be the same kind of  root. Yours are not the same kind.<br>
In a situation like this, replace the radicals with fractional exponents. This often provides a way to simplify when the radical form does not. Fourth roots are the same as an exponent of 1/4 and square roots are the same as an exponent of 1/2. Rewriting the radicals with fractional exponents we get:
{{{(6a^12*b^5)^(1/4) * (6a^12*b^5)^(1/2)}}}
Looking at your expression this way we can see that the bases of both factors are the same! This makes multiplying them very easy. The rule for exponents for this, {{{w^p * w^q = w^(p+q)}}}, tells us to add the exponents. Using this we get:
{{{(6a^12*b^5)^((1/4)+(1/2))}}}
To add the fractions we must have common denominators, of course:
{{{(6a^12*b^5)^((1/4)+(2/4))}}}
{{{(6a^12*b^5)^(3/4)}}} 
Now we can go back to radical form:
{{{root(4, (6a^12*b^5)^3)}}}
which simplifies to
{{{root(4, 216a^36*b^15)}}}
Last of all, we simplify the fourth root by finding power of 4 factors in the radicand, {{{216a^36*b^15}}}. There are no power of 4 factors in 216. But {{{a^36}}} is a power of 4 itself since the exponent is a multiple of 4. {{{b^15}}} is not a power of 4 itself since 15 is not a multiple of 4. But it does have a power of 4 factor: {{{b^15 = b^12*b^3}}}. So we can factor the radicand to show the power of 4 factors:
{{{root(4, 216a^36*b^12*b^3)}}}
At this point I like to use the Commutative property to rearrange the order so that the power of 4 factors are in front:
{{{root(4, a^36*b^12*216*b^3)}}}
Now we can use the property {{{root(a, p) * root(a, q) = root(a, p*q)}}} (from right to left) to split this radical of a product into a product of radicals. We want each power of 4 factor in their own radical. The factors that are not powers of 4 all go into one radical:
{{{root(4, a^36)*root(4, b^12)*root(4, 216*b^3)}}}
Now we can simplify the 4th roots of the powers of 4:
{{{root(4, (a^9)^4)*root(4, (b^3)^4)*root(4, 216*b^3)}}}
{{{abs(a^9)*b^3*root(4, 216*b^3)}}}
or
{{{a^8*abs(a)*b^3*root(4, 216*b^3)}}}
Either of these are your answer. (Note that the radical is at the end. This is the usual way to write terms with radicals. This is why I put the power of 4 factors in the front. One way or another they were going to end up in front anyway.)<br>
Many problems like this have some statement that tells you that the variables are positive (or at least not negative). If this statement was present and you just neglected to include it in your post, then your answer is just:
{{{a^9*b^3*root(4, 216*b^3)}}}<br>
Without this statement, the absolute value can become necessary. Let me take a moment to explain this need for absolute value and why it was used on {{{root(4, (a^9)^4)}}} but not on {{{root(4, (b^3)^4)}}}:
Even numbered roots, like 4th and square roots, are supposed to be positive or zero (IOW non-negative). So your original expression is a non-negative times a non-negative. A non-negative times a non-negative equals a non-negative. So your original expression, no matter what values "a" or "b" may have, will always be non-negative. Any simplified expression of your original expression should also be non-negative, no matter what.<br>
This is why we use the absolute value on {{{root(4, (a^9)^4)}}}. This should be non-negative and so should it's simplified {{{a^9}}}. But {{{a^9}}} would be negative is "a" was negative. Since "a" could be negative, we put absolute value on {{{a^9}}} to make sure it is non-negative. We can also use {{{a^8*abs(a)}}} instead of {{{abs(a^9)}}} because "a" to an even power like 8 must be non-negative and so it does not need the absolute value.<br>
We do not need an absolute value on the {{{b^3}}}. {{{root(4, (b^3)^4)}}} must be non-negative just like every 4th root. But for "b" we know that it cannot be negative. How? Look at the original expression:
{{{root(4, 6a^12*b^5) * sqrt(6a^12*b^5)}}}
We see that {{{b^5}}} is a factor in the radicand of both even-numbered roots. If "b" was negative, then {{{b^5}}} would be negative, too. And if {{{b^5}}} is negative then {{{216a^12*b^5}}} would be negative (since 216 is obviously positive and {{{a^12}}} is positive because any even-numbered power must be non-negative. But the radicands (expressions within a radical) of even-numbered roots cannot be negative! (You can't square something or raise something to the 4th power and get a negative result!) For this reason "b" cannot be a negative number. And if "b" cannot be negative neither will the {{{b^3}}} in the answer. This is why we do not need absolute value on the {{{b^3}}}. We already know that it must be non-negative.<br>
We cannot use the same logic to say that "a" cannot be negative. The original radicands have {{{a^12}}} in them. "a" to an even power will always be non-negative. So, unlike "b", we cannot say that "a" cannot be negative. It could be negative.<br>
This is why we must use absolute value on the {{{a^9}}} but not on the {{{b^3}}}. If it bothers you you could put absolute value on the whole thing:
{{{abs(a^9*b^3)*root(4, 216*b^3)}}}
But I think that either
{{{abs(a^9)*b^3*root(4, 216*b^3)}}}
or
{{{a^8*abs(a)*b^3*root(4, 216*b^3)}}}
would be the preferred form.