SOLUTION: Please help me solve this question in a understandable way for grade 10 curriculum! Given that 3a−1/2a=7, determine the value of 27a^3−1/8a^3 without solving for a.

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Question 1044562: Please help me solve this question in a understandable way for grade 10 curriculum!
Given that 3a−1/2a=7, determine the value of 27a^3−1/8a^3 without solving for a.
From the website, it gives that (3a−1/2a)(9a^2+3/2+1/4a^2)= 27a^3-1/8a^3, but how did they get the 3/2 from the middle?
Answers will be much appreciated!
Found 3 solutions by josgarithmetic, ikleyn, MathTherapy:
Answer by josgarithmetic(39617) About Me  (Show Source):
You can put this solution on YOUR website!
The equation which the website shows is based on the formula for Difference Of Cubes. You can find the formula yourself using polynomial division.
Something is missing in what the website tells you. You have found a mistake.

Look: x%5E3-y%5E3=%28x-y%29%28x%5E2%2Bxy%2By%5E2%29


Maybe I have not understood some of what YOU wrote?
Did you mean, 3a-%281%2F2%29a=7
or did you mean, 3a-1%2F%282a%29=7
?

The way that 3%2F2 becomes one of the terms in one of the polynomial sum-of-terms factors is therefore becoming more obvious.

Answer by ikleyn(52777) About Me  (Show Source):
You can put this solution on YOUR website!
.
Please help me solve this question in a understandable way for grade 10 curriculum!
Given that 3a - 1/(2a) = 7, determine the value of 27a^3 - 1/(8a^3) without solving for a.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 

You are given

3a+-+1%2F%282a%29 = 7.       (1)

Square both sides. You will get

9a%5E2+-+2%2A3%2A%281%2F2%29 + 1%2F%284a%5E2%29 = 49.

Or

9a%5E2+-+3 + 1%2F%284a%5E2%29 = 49.

Add 41%2F2 to both sides. You will get

9a%5E2+%2B+3%2F2 + 1%2F%284a%5E2%29 = 49 + 41%2F2 = 531%2F2.

Notice this one more time: 

9a%5E2+%2B+3%2F2 + 1%2F%284a%5E2%29 = 53.5.     (2)

Now

  27a%5E3 - 1%2F%288a%5E3%29 =      (apply this identity: x%5E3-y%5E3 = %28x-y%29%2A%28x%5E2+%2B+xy+%2B+y%5E2%29 !)

= %283a+-+1%2F%282a%29%29.(9a%5E2+%2B+3%2F2 + 1%2F%282a%29%5E2) =   (now recall that the first parentheses is 7 (given !), while the second is 53.5 due to (2) !)

= 7*53.5.

Please calculate the last product on your own.

Is it what you want/need ?

Done.

Why/how I was so smart?

Because I knew the content of the lesson
    - HOW TO evaluate expressions involving  %28x+%2B+1%2Fx%29,  %28x%5E2%2B1%2Fx%5E2%29  and  %28x%5E3%2B1%2Fx%5E3%29
in this site, so, I was prepared in advance.

Now, after reading my post and this lesson, you will be prepared too.

By the way, you have free of charge online textbook in "Algebra-I" in this site
    - ALGEBRA-I - YOUR ONLINE TEXTBOOK.


Answer by MathTherapy(10551) About Me  (Show Source):
You can put this solution on YOUR website!

Please help me solve this question in a understandable way for grade 10 curriculum!
Given that 3a−1/2a=7, determine the value of 27a^3−1/8a^3 without solving for a.
From the website, it gives that (3a−1/2a)(9a^2+3/2+1/4a^2)= 27a^3-1/8a^3, but how did they get the 3/2 from the middle?
Answers will be much appreciated!
3a+-+%281%2F2%29a+=+7

27a%5E3+-+%281%2F8%29a%5E3 =====> %283a%29%5E3+-+%28%281%2F2%29a%29%5E3  

%283a+-+%281%2F2%29a%29%289a%5E2+%2B+3a%5E2%2F2+%2B+%281%2F4%29a%5E2%29 <====== This is what you should've seen

The above reduces to: highlight_green%28301a%5E2%2F4%29