SOLUTION: 3 ∜ 24 multiplied by 5 ∜ 2 why is the answer 30 ∜3

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Question 395159: 3 ∜ 24
multiplied by
5 ∜ 2
why is the answer 30 ∜3
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
3%2Aroot%284%2C+24%29%2A5%2Aroot%284%2C+2%29
This is all multiplication. So we can use the Commutative and Associative Properties to rearrange the order and grouping in any way we choose:
%283%2A5%29%2A%28root%284%2C+24%29%2Aroot%284%2C+2%29%29
To multiply the radicals we use the property of all radicals: root%28a%2C+p%29%2Aroot%28a%2C+q%29+=+root%28a%2C+p%2Aq%29:
15%2Aroot%284%2C+24%2A2%29
or
15%2Aroot%284%2C+48%29
Just like fractions should be reduced, radicals should be simplified. Simplifying radicals involves finding factors of the radicand (the expression within a radical) that are powers of the type of root, if any. In this problem with its 4th root, we are looking for factors of 48 that are a power of 4. Since 2%5E4+=+16 and since 16 is a factor of 48, this radical will simplify. We start by writing the radicand in factored form:
15%2Aroot%284%2C+16%2A3%29
Then we use the property of radicals we used earlier. Only this time we use it in the other direction: to split a single radical into a product of radicals. By doing this we get the power of 4 factor into its own radical:
15%2Aroot%284%2C+16%29%2Aroot%284%2C+3%29
The 4th root of 16 is 2 so this becomes:
15%2A2%2Aroot%284%2C+3%29
which simplifies to
30%2Aroot%284%2C+3%29