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Multiplying Radicals of Different Roots - Concept
To simplify two radicals with different roots, we first rewrite the roots as rational exponents. Before the terms can be multiplied together, we change the exponents so they have a common denominator. By doing this, the bases now have the same roots and their terms can be multiplied together. Next, we write the problem using root symbols and then simplify.
So we know how to multiply square roots together when we have the same index, the same root that we're dealing with. What we don't know is how to multiply them when we have a different root. So that's what we're going to talk about right now.
So if we have the square root of 3 times the square root of 5. They're both square roots, we can just combine our terms and we end up with the square root 15. Okay? That's easy enough. What we don't really know how to deal with is when our roots are different. So what I have here is a cube root and a square root, okay? As is we can't combine these because we're dealing with different roots. But there is a way to manipulate these to make them be able to be combined. And how I always do this is to rewrite my roots as exponents, okay? So turn this into 2 to the one third times 3 to the one half. Okay. And remember that when we're dealing with the fraction of exponents is power over root. In order to multiply our radicals together, our roots need to be the same. So we somehow need to manipulate these 2 roots, the 3 and the squared, the 3 and the 2 to be the same root, okay? So think about what our least common multiple is. 2 and 3, 6. Okay? So we want to rewrite these powers both with a root with a denominator of 6. So 6, 2 you get a 6. We just need to multiply that by 2 over 2, so we end up with 2 over 6 and then 3, need to make one half with the denominator 6 so that's just becomes 3 over 6. Okay. So what we really have right now then is the sixth root of 2 squared times the sixth root of 3 to the third. Okay? So we didn't change our problem at all but we just changed our exponent to be a little but bigger fraction. That's perfectly fine. And now we have the same roots, so we can multiply leaving us with the sixth root of 2 squared times 3 cubed. Okay. Often times these numbers are going to be pretty ugly and pretty big, so you sometimes will be able to just leave it like this. 2 squared and 3 cubed aren't that big of numbers. 2 squared is 4, 3 squared is 27, 4 times 27 is I believe 108. So this becomes the sixth root of 108.
Just a little side note, you don't necessarily have to go from rewriting it from your fraction exponents to your radicals. It often times it helps people see exactly what they have so seeing that you have the same roots you can multiply but if you're comfortable you can just go from this step right down to here as well. That's perfectly fine.
So whenever you are multiplying radicals with different indices, different roots, you always need to make your roots the same by doing and you do that by just changing your fraction to be a [IB] common denominator.