Question

Q13. If x + 1/x = 5, find the value of x^3+ 1/x^3
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Answer 1

Given:-

$\rm x+\dfrac{1}{x} = 5$

To find:-

$\rm x^3+\dfrac{1}{x^3}$

Solution:-

$\rm 5= x+\dfrac{1}{x}$

Cubing both the sides, we get :

$\rm\dashrightarrow (5)^3=\Bigg ( x+\dfrac{1}{x} \Bigg )^3$

$\rm{Since, (a+b)^3=a^3+b^3+3ab(a+b) , \therefore }$

$\rm\dashrightarrow (5)^3=(x)^3+\Bigg (\dfrac{1}{x}\Bigg )^3+3(x)\Bigg (\dfrac{1}{x}\Bigg ) \Bigg ( x +\dfrac{1}{x} \Bigg ) \\ \\\dashrightarrow \rm 125 = x^3+\dfrac{1}{x^3}+ 3(5) \qquad ...[since, x+\dfrac{1}{x}=5] \\ \\\dashrightarrow \rm 125= x^3+\dfrac{1}{x^3} +15$

$\rm\dashrightarrow x^3+\dfrac{1}{x^3} = 125-15$

$\rm\dashrightarrow x^3+\dfrac{1}{x^3} = 110$

Hence,

$\qquad\qquad\bigstar\boxed{\bf x^3+\dfrac{1}{x^3} = 110}\bigstar$