SOLUTION: Why is it NOT necessary for two radical expressions have the same index if they are to be multiplied? Provide your own example including correct answer.

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Question 1134898: Why is it NOT necessary for two radical expressions have the same index if they are to be multiplied? Provide your own example including correct answer.
Answer by MathLover1(20849) About Me  (Show Source):
You can put this solution on YOUR website!

One way to look at this is to take two radicals: sqrt%28a%29 and root%283%2Cb%29, which differing indices.
But we can introduce the LCM+of 2 and 3 which is 6 and write
sqrt%28a%29=root%286%2Ca%5E3%29 and root%283%2Cb%29=root%286%2Cb%5E2%29 where the number 2 and 3 is the index.
This introduces a common radical - 6th root.
Now we multiply under the same+index:
.
So in the general case we need the LCM of the indices as the+common radical and then we use the power indices under the common radical.
So it is not+necessary for 2 radical expressions to have the same+index in order to multiply them.
NUMERICAL EXAMPLE:
sqrt%28121%29 and root%283%2C27%29
sqrt%28121%29=11 and root%283%2C27%29=3
We know the answer is 11%2A3=33.
But let's see if we get the same result by putting a=121 and+b=27 in the formula above.
sqrt%28121%29%2Aroot%283%2C27%29
=root%286%2C%2811%5E2%29%5E3%29%2Aroot%286%2C%283%5E3%29%5E2%29
=root%286%2C177561%2A729%29%7D%7D%0D%0A%0D%0A=%7B%7B%7B11%2A3
=33
So the formula works.