Question

find the value of x^4+1/x^4 when x-1/x=5​

Answer 1

Answer:

$\huge{ \bold{ \boxed{ \pink{ \star{x - \frac{1}{x} = 5}}}}}$

$\huge{ \bold{ \boxed{ \pink{ \star{ {x}^{2} + {( \frac{1}{x} })^{2} + 2 = 25}}}}}$

$\huge{ \bold{ \boxed{ \pink{ \star{ {x}^{2} + \frac{1}{ {x}^{2} } = 23}}}}}$

$\huge{ \bold{ \boxed{ \green{ \star{ {x}^{4} + \frac{1}{ {x}^{4} } + 2 = 529}}}}}$

$\huge{ \bold{ \boxed{ \blue{ \star{ {x}^{4} + \frac{1}{ {x}^{4} } = 527}}}}}$

Answer 2

$\huge{\blue{\bold{\boxed{\ulcorner{\star\:Answer\: \star}\urcorner}}}}$

Given x + 1/x = 5

Squaring, (x + 1/x)² = 5² = 25

Or, expanding x² + 2.x.1/x + 1/x² = 25

Or, x² + 2 + 1/x² = 25

Transposing 2 to right and simplifying

x² + 1/x² = 23…….(1)

Now

x⁴ + 1/x⁴ = [(x²)² + 2.x². 1/x² + (1/x²)²] - 2.x². 1/x² = (x² + 1/x²)² - 2

= 23² - 2 [Substituting for x² + 1/x² from (1)]

= 529 - 2 = 527 (Proved)

Method 2:

x + 1/x = 5 …….(1)

Multiplying (1) through out by x and rearranging,

x² - 5x + 1 = 0

The equation above being a quadratic in x furnishes two solutions which

are given by

x = [5±√(5² - 4.1.1)]/2 = (5±√21)/2

∴ x = (5+√21)/2 and x = (5-√21)/2

x = (5 + √21)/2

∴ x⁴ = (5 + √21)⁴/2⁴ = (5 + 4.58257569495)⁴/16 = (9.58257569495)⁴/16

= 8431.96963932/16 = 526.998102457

and 1/x⁴ = .00189754003

∴ x⁴ + 1/x⁴ = 526.998102457 + 0.00189754003 = 526.999999997 =527 if exact value of √21 is taken in the calculation.

x = (5 - √21)/2

∴ x⁴ = (5 - √21)⁴/2⁴ = (5 - 4.58257569495)⁴/16 = (.41742430505)⁴/16

= .03036064062/16 = .00189754003

and 1/x⁴ = 526.998105014

∴ x⁴ + 1/x⁴ = .00189754003 + 526.998105014 = 527.000002554 = 527 (approx.)

Hence, x⁴ + 1/x⁴ = 527 (Proved)