If x-1 x =5, find the value of x^3-1 x^3 .

Question

If $\text{x}-\frac{1}{\text{x}}=5,$ find the value of $\text{x}^3-\frac{1}{\text{x}^3}.$

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Answer

Given,If $\text{x}-\frac{1}{\text{x}}=5$ We know that, $(a - b)^3 = a^3 - b^3 - 3ab(a - b)$ ...(1)
Substitute $\text{x}-\frac{1}{\text{x}}=5$ in eq(1)$\Rightarrow\Big(\text{x}-\frac{1}{\text{x}}\Big)^3=\text{x}^3-\frac{1}{\text{x}^3}-3\Big(\text{x}\times\frac{1}{\text{x}}\Big)\Big(\text{x}-\frac{1}{\text{x}}\Big)$
$\Rightarrow5^3=\text{x}^3-\frac{1}{\text{x}^3}-3\Big(\text{x}-\frac{1}{\text{x}}\Big)$
$\Rightarrow125=\text{x}^3-\frac{1}{\text{x}^3}-(3\times5)$
$\Rightarrow125=\text{x}^3-\frac{1}{\text{x}^3}-15$
$\Rightarrow125+15=\text{x}^3-\frac{1}{\text{x}^3}$
$\Rightarrow\text{x}^3-\frac{1}{\text{x}^3}=140$
Hence, the result is $\text{x}^3-\frac{1}{\text{x}^3}=140.$

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