Question

If [x - 1/x] = 4, find the values of (i) [x²+ 1/x²] (ii) [x⁴ + 1/x⁴].​

Answer 1

Answer:

18 & 322

Step-by-step explanation:

Square on both sides:

→ (x - 1/x)² = 4²

→ (x)² + (1/x)² - 2(x)(1/x) = 16

→ x² + 1/x² - 2(x * 1/x) = 16

→ x² + 1/x² - 2(1) = 16 {x*1/x = 1}

→ x² + 1/x² = 16 + 2 = 18

Square on both sides again:

→ (x² + 1/x²)² = 18²

→ (x²)² + (1/x²)² + 2(x²)(1/x²) = 324

→ x⁴ + 1/x⁴ + 2(1) = 324 {(x²)²=x⁴}

→ x⁴ + 1/x⁴ =324 - 2 = 322

Answer 2

$x - \frac{1}{x} = 4 \\ \\ ( {x - \frac{1}{x} })^{2} = 16 \\ \\ {x}^{2} + \frac{1}{ {x}^{2} } - 2 = 16 \\ \\ {x}^{2} + \frac{1}{ {x}^{2} } = 18 \\ \\ ( { {x}^{2} + \frac{1}{ {x}^{2} } })^{2} = 324 \\ \\ {x}^{4} + \frac{1}{ {x}^{4} } + 2= 324 \\ \\ {x}^{4} + \frac{1}{ {x}^{4} } = 322$