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Answers
Answer is:
243.
Solution is as follows:
The desired difference looks like (a^3) - (b^3) with a = (3x) and b = (1/2x).
The first equation (2x) - (1/3x) looks like (a - b) but the terms seem different. To get (3x) - (1/2x) from first equation we can multiply the first equation on both sides by (3/2):
(3/2)[(2x) - (1/3x)] = (3/2) * 4
=> (3x) - (1/2x) = 6
Now that we have (a - b), we can compute (a^3) - (b^3) using the identity:
(a^3) - (b^3) = (a - b)(a^2 + ab + b^2) = A * B
We already know:
A: (3x) - (1/2x) = 6
Now we need to compute: (a^2 + ab + b^2).
To get this we can square (a - b) and add (3ab) to both sides because:
(a - b)^2 = a^2 - 2ab + b^2
Adding (3ab) to both sides gets us:
(a - b)^2 + (3ab) = a^2 + ab + b^2
Starting off with (a - b)^2:
[(3x)-(1/2x)]^2 = 6^2
=> (9x^2) - (3) + (1/4x^2) = 36
Adding 3ab which is equal to 3*(3x)*(1/2x) = 9/2 to both sides results in:
=> B: (9x^2) + (3/2) + (1/4x^2) = (81/2)
Multiplying A with B gives us
(27x^3) - (1/8x^3) = 6 * (81/2) = 243
Therefore,
(27x^3) - (1/8x^3) = 243
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