Math, asked by nfkhan115, 2 months ago

if x² - 1/x²= 5 find the value of (i) x²+1/x² (ii)x⁴+ 1/x⁴ (iii) x³ - 1/x³​

Answers

Answered by amansharma264
14

EXPLANATION.

⇒ (x² - 1/x²) = 5.

To find the value of,

(1) = x² + 1/x².

⇒ (x² - 1/x²) = 5.

Squaring on both sides, we get.

⇒ (x - 1/x)² = (5)².

⇒ (x² + 1/x² - 2(x)(1/x)) = 25.

⇒ x² + 1/x² - 2 = 25.

⇒ x² + 1/x² = 25 + 2.

⇒ x² + 1/x² = 27.

(2) = x⁴ + 1/x⁴.

As we know that,

⇒ x² + 1/x² = 25.

Squaring on both sides, we get.

⇒ (x² + 1/x²)² = (25)².

⇒ (x⁴ + 1/x⁴ + 2(x²)(1/x²)) = 625.

⇒ x⁴ + 1/x⁴ + 2 = 625.

⇒ x⁴ + 1/x⁴ = 625 - 2.

⇒ x⁴ + 1/x⁴ = 623.

(3) = x³ - 1/x³.

As we know that,

⇒ (x - 1/x)² = x² + 1/x² - 2(x)(1/x).

⇒ (x - 1/x)² = x² + 1/x² - 2.

Put the value of x² + 1/x² in equation, we get.

⇒ (x - 1/x)² = 27 - 2.

⇒ (x - 1/x)² = 25.

⇒ (x - 1/x)² = (5)².

⇒ (x - 1/x) = 5.

We get the value of = x - 1/x = 5.

Now, we can cube on both sides, we get.

⇒ (x - 1/x)³ = (5)³.

⇒ (x³ - 3(x²)(1/x) + 3(x)(1/x²) - 1/x³) = 125.

⇒ x³ - 3x + 3/x - 1/x³ = 125.

⇒ x³ - 1/x³ - 3(x - 1/x) = 125.

⇒ x³ - 1/x³ - 3(5) = 125.

⇒ x³ - 1/x³ - 15 = 125.

⇒ x³ - 1/x³ = 125 + 15.

⇒ x³ - 1/x³ = 140.

Answered by gurmanpreet1023
6

Answer:

Given :

x + 1/x = 3

To find :

value of x² + 1/x², x³ + 1/x³, and x⁴+1/x⁴

Solution :

We have x + 1/x = 3 ……..(1)

On squaring eq 1 both sides,

(x + 1/x)² = 3²

By Using Identity : (a + b)² = a² + b² + 2ab

x² + 1/x² + 2 x × 1/x = 9

x² + 1/x² + 2 = 9

x² + 1/x² = 9 - 2

x² + 1/x² = 7 ………….(2)

On squaring eq 2 both sides,

(x² +1/x² )² = 7²

(x²)² + (1/x²)² + 2 x² × 1/x² =7²

x⁴ + 1/x⁴ + 2 = 49

x⁴ + 1/x⁴ = 49 - 2

x⁴ + 1/x⁴ = 47

On Cubing eq 1 both sides :

(x + 1/x)³ = 3³

By Using Identity : (a + b)³ = a³ + b³ + 3ab(a + b)

(x)³ + (1/x)³ + 3 × x× 1/x (x + 1/x) = 27

x³ + 1/x³ + 3(x + 1/x) = 27

x³ + 1/x³ + 3(3) = 27

x³ + 1/x³ + 9 = 27

x³ + 1/x³ = 27 - 9

x³ + 1/x³ = 18

Hence the value of the value of x² + 1/x² is 7 , x³ + 1/x³ is 18 & x⁴ + 1/x⁴ is 47.

HOPE THIS ANSWER WILL HELP YOU…..

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