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Multiplying Radicals of Different Roots - Problem 1
Multiplying radicals with different roots; so what we have to do whenever we're multiplying radicals with different roots is somehow manipulate them to make the same roots out of our each term. What we have behind me is a product of three radicals and there is a square root, a fourth root and then third root. We want to somehow combine those all together.
Whenever I'm dealing with a problem like this, the first thing I always do is take them from radical form and write them as an exponent okay? So the square root of 7 goes into 7 to the 1/2, the fourth root goes to 2 and one fourth and the cube root goes to 3 to the one-third.
Okay so from here what we need to do is somehow make our roots all the same and remember that when we're dealing with fractional exponents, the root is the denominator, so we want the 2, the 4 and the 3 to all be the same. Think of all these common multiples, so these common multiples are 3 numbers that are going to be 12, so we need to make our denominator for each exponent to be 12.
So that becomes 7 goes to 6 over 12, 2 goes to 3 over 12 and 3 goes to 4 over 12. So now we have the twelfth root of everything okay? Power of a root, these are all the twelfth roots. Once we have the roots the same, we can just multiply and end up with the twelfth root of 7 to the sixth times 2 to the third, times 3 to the fourth.
This is going to be a master of number, so in generally I'd probably just say you can leave it like this, if you have a calculator you can always plug it in and see what turns out, but it's probably going to be a ridiculously large number.
So what we did is basically taking our radicals, putting them in the exponent form, getting a same denominator so what we're doing is we're getting the same root for each term, once we have the same roots we can just multiply through.