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Every radical has an index which indicates which type of root it is. Since we are trying to get this entire expression into a single radical, we need to have all of the expression become the same kind of root. The type of root, indicated by the index of the radical, is also indicated by the denominator of a fractional exponent. So to make our expression all be the same kind of root, we need for the exponents to have the same denominator!
We start by giving the middle exponent a denominator:
Now what denominator can we make all three denominators become? (This is just like we were going to add the fractions. What's the lowest common denominator?) With denominators of 2,1 and 8 it should not be hard to figure out that 8 is the lowest common denominator. So the third exponent is fine. We just have to change the other two to have denominators of 8:
Next we use the property. Usually we use this property to go from right to left, to raise a product to a power. But here, we are going to use it in reverse:
If you're having trouble seeing how we got this, look at going from
to
We just went backwards.
We are now ready to write our radical expression. The exponent of 1/8 tells us to write an 8th root radical:
This may be an acceptable answer. But the expression will simplify:
(It shouldn't be a surprise that ended up outside the radical. Its exponent was not a fraction so it never really belonged in a radical to begin with.)
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