SOLUTION: (Quartic root sign over 18)�(square root 12)

Question 718191: (Quartic root sign over 18)�(square root 12)
Answer by jsmallt9(3758) (Show Source): You can put this solution on YOUR website!

There is a perfect square factor in 12. So the square root will simplify. Since simplifying the square root will make the rest of the problem a little easier, I will start with that:

Roots of different types cannot be multiplied. But if we can find a way to change one or both of theses roots so that we end up with the same type of root, then we could multiply.

The "trick" with this kind of problem is to understand and take advantage of the relationship between radicals and fractional exponents. Let's start by rewriting these radicals with fractional exponents:

In the fractional exponents the denominator indicates the type of root. We want the types of these two roots to be the same. In other words, we want the fractions to have the same denominator! So we find the lowest common denominator (LCD). In this case, it should be easy to see that the LCD will be 4. So the first fraction is just fine as it is. But we need to change 1/2 to 2/4:

Now that the types of the roots are the same we will return to radicals:

Simplifying the 3 squared:

We can now multiply the roots:

Since there are no power of 4 factors in 162, the root will not simplify. This is the simplified answer.

P.S. If we had not simplified at the beginning we would have ended up with which we would then have to try to simplify. It does have a power of 4 factor, 16, so it does simplify (to the answer we got above). But I hope you agree that simplifying at the beginning was much easier than simplifying at the end.
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