SOLUTION: simplify this expression: 2*(cube root of 3)+ 4*(cube root of 24)-(cube root of 81) please show steps and explain

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Question 217793: simplify this expression:
2*(cube root of 3)+ 4*(cube root of 24)-(cube root of 81)
please show steps and explain
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
2%2Aroot%283%2C3%29%2B4%2Aroot%283%2C24%29-root%283%2C81%29
First we simplify each cube root. To simplify cube roots, try to find perfect cube factors (other than 1) in the radicand (the number inside the radical).

The first radicand has no perfect cube factors. The second and third radicands, however, do have perfect cube factors: 8 and 27 respectively. We can then rewrite them:
2%2Aroot%283%2C3%29%2B4%2Aroot%283%2C8%2A3%29-root%283%2C27%2A3%29
Now we can use the property of roots, root%28a%2C+x%2Ay%29+=+root%28a%2C+x%29%2Aroot%28a%2Cy%29,
to separate the perfect cube factors into their own cube roots:

Since root%283%2C8%29+=+2 and root%283%2C+27%29+=+3 we have:
2%2Aroot%283%2C3%29%2B4%2A2%2Aroot%283%2C3%29-3%2Aroot%283%2C3%29
Simplifying the middle term:
2%2Aroot%283%2C3%29%2B8%2Aroot%283%2C3%29-3%2Aroot%283%2C3%29
Now we have simplified the individual cube roots. Next we try to simplify the entire expression. Can we add and subtract these terms? Answer: yes. They are like terms. Just like 2x+8x-3x are like terms so are these cube roots. If you have trouble seeing this, try using a variable. Let x=root%283%2C3%29. Then our equation becomes 2x+8x-3x! (On the other hand, 2%2Aroot%283%2C4%29%2B8%2Aroot%283%2C6%29 are not like terms because the radicands are different. And 2%2Aroot%283%2C+5%29+%2B+8%2Aroot%284%2C5%29 are not like terms because they are different roots.)


Since no more simplifying can be done, we are finished.