SOLUTION: what is the square root 3 of 54 - square root 3 of 128

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Question 271344: what is the square root 3 of 54 - square root 3 of 128
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
First of all, root%283%2C+54%29 means "cube root of 54".

root%283%2C+54%29+-+root%283%2C+128%29
These are not like terms so we cannot subtract them as they are. But, as with all radicals, we should try to simplify them. Simplifying cube roots involves finding perfect cube factors of the radicands (the expression within the radical), if any.
54 = 27*2 and 27 is 3 cubed.
128 = 64*2 and 64 is 4 cubed.
So we can rewrite the expression as:
root%283%2C+27%2A2%29+-+root%283%2C+64%2A2%29
Now we can use the property of radicals, root%28a%2C+p%2Aq%29+=+root%28a%2C+p%29%2Aroot%28a%2C+q%29 to split away the perfect cube factors into their own radicals:
root%283%2C+27%29%2Aroot%283%2C+2%29+-+root%283%2C+64%29%2Aroot%283%2C+2%29
We can now simplify the cube roots of the perfect cubes:
3%2Aroot%283%2C+2%29+-+4%2Aroot%283%2C+2%29
At this point, not only have we simplified the cube roots, we also have like terms which can be subtracted! Just like 3x - 4x = -x,
3%2Aroot%283%2C+2%29+-+4%2Aroot%283%2C+2%29+=+-root%283%2C+2%29