SOLUTION: Prove a^3-64/a^2-16=a^2+4a+16/a+4

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Question 111367: Prove a^3-64/a^2-16=a^2+4a+16/a+4
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
For this problem we need to use the difference of squares and the difference of cubes formulas.

Remember the difference of squares formula is:
x%5E2-y%5E2=%28x%2By%29%28x-y%29

and the difference of cubes formula is:
x%5E3-y%5E3=%28x-y%29%28x%5E2%2Bxy%2By%5E2%29


So rewrite the numerator a%5E3-64 as %28a%29%5E3-%284%29%5E3 notice how x=a and y=b for the formula x%5E2-y%5E2=%28x%2By%29%28x-y%29. Now factor %28a%29%5E3-%284%29%5E3 to %28a-4%29%28a%5E2%2B4a%2B16%29

So our expression now looks like this: %28a-4%29%28a%5E2%2B4a%2B16%29%2F%28a%5E2-16%29

Now factor the denominator using the difference of squares

%28a-4%29%28a%5E2%2B4a%2B16%29%2F%28%28a%2B4%29%28a-4%29%29


Cancel like terms

cross%28%28a-4%29%29%28a%5E2%2B4a%2B16%29%2F%28%28a%2B4%29cross%28%28a-4%29%29%29


Simplify

%28a%5E2%2B4a%2B16%29%2F%28a%2B4%29

So this shows that %28a%5E3-64%29%2F%28a%5E2-16%29 simplifies to %28a%5E2%2B4a%2B16%29%2F%28a%2B4%29