SOLUTION: Evaluate a^3 + 1/a^3 if a + 1/a = 6.

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Question 1066053: Evaluate a^3 + 1/a^3 if a + 1/a = 6.
Found 2 solutions by ikleyn, MathTherapy:
Answer by ikleyn(52781) About Me  (Show Source):
You can put this solution on YOUR website!
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Evaluate a^3 + 1/a^3 if a + 1/a = 6.
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Let  a+%2B+1%2Fa = 6.    (1)

Cube both sides of the equation (1). You will get

a%5E3 + 3%2Aa%5E2%2A%281%2Fa%29 + 3%2Aa%2A%281%2Fa%5E2%29 + 1%2Fa%5E3 = 6%5E3 =  216   (after using the formula for the cube of a sum %28x%2By%29%5E3 = x%5E3+%2B+3%2Ax%5E2%2Ay+%2B+3%2Ax%2Ay%5E2+%2B+y%5E3).

Now, cancel the factors a and 1%2Fa. You will get

a%5E3 + 3%2Aa + 3%2A%281%2Fa%29 + 1%2Fa%5E3 = 216,    or, after grouping the terms,

a%5E3 + 3%2A%28a+%2B+%281%2Fa%29%29 + 1%2Fa%5E3 = 216.

Replace a+%2B+1%2Fa by 6, in accordance with the condition. You will get

a%5E3 + 3%2A6 + 1%2Fa%5E3 = 216    or

a%5E3 + 1%2Fa%5E3 = 216 - 3%2A6 = 216+-+18 = 198.

Thus   a%5E3 + 1%2Fa%5E3 = 198%29.

Answer.   a%5E3 + 1%2Fa%5E3 = 198.

See the lesson
    - HOW TO evaluate expressions involving  %28x+%2B+1%2Fx%29,  %28x%5E2%2B1%2Fx%5E2%29,  %28x%5E3+%2B+1%2Fx%5E3%29  and  %28x%5E5%2B1%2Fx%5E5%29
in this site.

Also, you have this free of charge online textbook in ALGEBRA-I in this site
    - ALGEBRA-I - YOUR ONLINE TEXTBOOK.

The referred lesson is the part of this online textbook under the topic "Evaluation, substitution".



Answer by MathTherapy(10552) About Me  (Show Source):
You can put this solution on YOUR website!

Evaluate a^3 + 1/a^3 if a + 1/a = 6.
a%5E3+%2B+1%2Fa%5E3         			a+%2B+1%2Fa+=+6

%28a+%2B+1%2Fa%29%5E2+=+6%5E2 ------ Squaring the equation: a+%2B+1%2Fa+=+6
a%5E2+%2B+2+%2B+1%2Fa%5E2+=+36
a%5E2+%2B+1%2Fa%5E2+=+34

a%5E3+%2B+1%2Fa%5E3
%28a+%2B+1%2Fa%29%28a%5E2+-+1+%2B+1%2Fa%5E2%29 ------ Factoring a%5E3+%2B+1%2Fa%5E3
%28a+%2B+1%2Fa%29%28a%5E2+%2B+1%2Fa%5E2+-+1%29

6(34 - 1) ------- Substituting matrix%281%2C7%2C+6%2C+for%2C+a+%2B+1%2Fa%2C+and%2C+34%2C+for%2C+a%5E2+%2B+1%2Fa%5E2%29