Question

) If (x - 1/x) = 8, find the value of (x² + 1/x²) and (x⁴ + 1/x⁴).​

Answer 1

Step-by-step explanation:

the formula of A2+b2 formula used

Answer 2

Given:

• $\small{\tt{x-\frac{1}{x}~=~8}}$

To Find:

• $\small{\tt{x^2+\frac{1}{x^2}}}$

• $\small{\tt{x^4+\frac{1}{x^4}}}$

Solution:

Answer of questions 1st:-

$\small{\sf{x-\frac{1}{x}~=~8}}$

[Squaring both side]

$\implies\small{\sf\bigg({x-\frac{1}{x}\bigg)}^2~=~8^2}$

$\implies\small{\sf{x^2+\frac{1}{x^2}-2\times x \times \frac{1}{x}~=~64}}$

$\implies\small{\sf{x^2+\frac{1}{x^2}-2\times{\cancel{x}}\times \frac{1}{\cancel{x}}~=~64}}$

$\implies\small{\sf{x^2+\frac{1}{x^2}~=~64+2}}$

$\implies\small{\boxed{\bf{x^2+\frac{1}{x^2}~=~66}}}$

Answer of questions 2nd:-

$\implies\small{\sf{x^2+\frac{1}{x^2}~=~66}}$

[Squaring both side]

$\implies\small{\sf\bigg({x^2+\frac{1}{x^2}\bigg)}^2~=~66^2}$

$\implies\small{\sf{x^4+\frac{1}{x^4}+2\times x^2 \times \frac{1}{x^2}~=~66^2}}$

$\implies\small{\sf{x^4+\frac{1}{x^4}+2\times{\cancel{x^2}}\times \frac{1}{\cancel{x^2}}~=~66^2}}$

$\implies\small{\sf{x^4+\frac{1}{x^4} = 66^2 - 2}}$

$\implies\small{\sf{x^4+\frac{1}{x^4}~=~4356 - 2}}$

$\implies\small{\boxed{\sf{x^4+\frac{1}{x^4}~=~4354}}}$

Concept used:

• (a+b)² = a² + b² + 2ab
• (a-b)² = a² + b² - 2ab