SOLUTION: cube root of 2(cube root of 4 subtract 2 times the cube root of 32)

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Question 711333: cube root of 2(cube root of 4 subtract 2 times the cube root of 32)
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
If
root%283%2C+2%29%28root%283%2C+4%29-2%2Aroot%283%2C+32%29%29
is the correct expression then please post it as:
(cube root of 2)((cube root of 4) subtract (2 times the cube root of 32))
The extra parentheses make it clear. Without them
cube root of 2(cube root of 4 subtract 2 times the cube root of 32)
could be interpreted as:
root%283%2C+2%2Aroot%283%2C+4+-+2%2Aroot%283%2C+32%29%29%29

If
root%283%2C+2%29%28root%283%2C+4%29-2%2Aroot%283%2C+32%29%29
is not correct then you will have to re-post your question because I'm going to simplify this expression.

We could start with either using the Distributive Property to multiply by the root%283%2C+2%29 or we could simplify root%283%2C+32%29 since 8 is a factor of 32 and 8 is a perfect cube (8+=+2%5E3). Looking ahead I can see that using the Distributive Property first will make the rest much easier. So that is how I will start:
root%283%2C+2%29%2Aroot%283%2C+4%29-root%283%2C+2%29%2A2%2Aroot%283%2C+32%29%29
Multiplying the roots together, using the root%28a%2C+p%29%2Aroot%28a%2C+q%29+=+root%28a%2C+p%2Aq%29 property of radicals we get:
root%283%2C+2%2A4%29-2%2Aroot%283%2C+2%2A32%29%29
Simplifying inside the radicals we get:
root%283%2C+8%29-2%2Aroot%283%2C+64%29%29
8, as was mentioned earlier, is a perfect cube. And so is 64! 64+=+4%5E3. So both cube roots simplify:
2-2%2A4
Continuing to simplify:
2-8
-6