**If x**^{2} + 1/4x^{2} = 8, find x^{3} + 1/8x^{3}.

^{2}+ 1/4x

^{2}= 8, find x

^{3}+ 1/8x

^{3}.

**Answer :**

We know that

(x + 1/2x)^{2} = x^{2} + (1/2x)^{2} + 2x (1/2x)

It can be written as

(x + 1/2x)^{2} = x^{2} + 1/4x^{2} + 1

Substituting the values

(x + 1/2x)^{2} = 8 + 1 = 9

So we get

x + 1/2x = ± √9 = ± 3

Here

x^{3} + 1/8x^{3} = x^{3} + (1/2x)^{3}

We know that

x^{3} + 1/8x^{3} = (x + 1/2x)^{3} – 3x (1/2x) (x + 1/2x)

Substituting the values

x^{3} + 1/8x^{3} = (± 3)^{3} – 3/2 (± 3)

By further calculation

x^{3} + 1/8x^{3} = ± (27 – 9/2)

Taking LCM

x^{3} + 1/8x^{3} = ± (54 – 9)/ 2

x^{3} + 1/8x^{3} = ± 45/2 = ± 22 ½

Therefore, x^{3} + 1/8x^{3} = ± 22 ½.

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