If x+1 x =7, then x^3-1 x^3 = - Vidyadip

MCQ

If $x+\frac{1}{x}=7$, then $x^3-\frac{1}{x^3}=$

A
$9 \sqrt{5}$
✓
$144 \sqrt{5}$
C
$135 \sqrt{5}$
D
$\sqrt{5}$

✓

Answer

Correct option: B.

$144 \sqrt{5}$

(b)
We have,$x+\frac{1}{x}=7$
$\Rightarrow \quad\left(x+\frac{1}{x}\right)^2=7^2 \Rightarrow x^2+\frac{1}{x^2}+2=49 \Rightarrow x^2+\frac{1}{x^2}=47$
$\Rightarrow \quad x^2+\frac{1}{x^2}-2=45 \Rightarrow\left(x-\frac{1}{x}\right)^2=(3 \sqrt{5})^2 \Rightarrow x-\frac{1}{x}=3 \sqrt{5}$
$\therefore \quad x^3-\frac{1}{x^3}=\left(x-\frac{1}{x}\right)\left(x^2+\frac{1}{x^2}+1\right)=3 \sqrt{5}(47+1)=144 \sqrt{5}$

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