Question

If x-1/x=√5 then find the value of x³+1/x³ please help​

Answer 1

Answer:

$.$

Step-by-step explanation:

Given -

x + 1/x = 5

To Find -

Value of x³ + 1/x³

Now,

→ x + 1/x = 5

Cubing both sides :-

→ (x + 1/x)³ = (5)³

→ x³ + 1/x³ + 3x²/x + 3x/x² = 125

→ x³ + 1/x³ = 125 - 3x - 3/x

→ x³ + 1/x³ = 125 - 3(x + 1/x)

→ x³ + 1/x³ = 125 - 3(5)

→ x³ + 1/x³ = 125 - 15

→ x³ + 1/x³ = 110

Hence,

The value of x³ + 1/x³ is 110

Answer 2

$\maltese \:\:\underline {\pmb{\bf Given }}\:\::$

$\qquad \star \:\:\underline {\boxed {\pmb{\sf x - \dfrac{1}{x} \:=\:\sqrt{5} \:\:}}}\\\\\qquad\bigstar \:\:\underline{\pmb{\sf By \:\:\:Squaring\:\: \:both\:\: \:sides\:\: \:}}$

$\qquad :\implies \sf x - \dfrac{1}{x} \:=\:\sqrt{5} \\\\\\ \qquad :\implies \sf \bigg( x - \dfrac{1}{x} \bigg)^2 \:=\:\big( \sqrt{5}\big)^2 \:\:\\\\\\ \qquad :\implies \sf x^2 + \dfrac{1}{x^2} \:-\:2 \:\bigg( x \:\times \dfrac{1}{x}\bigg) \:\:=\: 5 \\\\\\\qquad :\implies \sf x^2 + \dfrac{1}{x^2} \:-\:2 \: \:\:=\: 5 \\\\\\ \qquad :\implies \sf x^2 + \dfrac{1}{x^2} \:\: \:\:=\: 5 + 2 \\\\\\\qquad :\implies \sf x^2 + \dfrac{1}{x^2} \: \: \:\:=\: 7$

Now ,

$\qquad :\implies \sf x^3 + \dfrac{1}{x^3} \\\\\\ \qquad :\implies \sf \bigg( x - \dfrac{1}{x} \bigg) \bigg[ x^2 + \dfrac{1}{x^2} + \bigg( x \times \dfrac{1}{x}\bigg) \bigg] \\\\\\ \qquad :\implies \sf \bigg( x - \dfrac{1}{x} \bigg) \bigg[ x^2 + \dfrac{1}{x^2} + 1 \bigg] \\\\\\ \qquad :\implies \sf \big( \sqrt{5} \big) \big( 7 + 1 \big) \\\\\\ \qquad :\implies \sf \big( \sqrt{5} \big) \big( 8 \big) \\\\\\ \qquad :\implies \sf x^3 + \dfrac{1}{x^3} = 8 \sqrt{5}$