Question

if x²-1/x²=4 find. the value of x³-1/x³​

Answer 1

Step-by-step explanation:

We know that

(a + b)² = a² + b² + 2ab

Therefore

(x + 1/x)² = x² + 1/x² + 2×x×1/x

or, (x + 1/x)² = 7 + 2 . . . . . . . . . . . . . . { x² + 1/x² = 7 is given}

or, (x + 1/x)² = 9

or, x + 1/x = √9

So, x + 1/x = ±3

We also know that

a³ + b³ + 3ab(a + b) = (a + b)³

Therefore

x³ + 1/x³ + 3 × x × 1/x (x + 1/x) = (x + 1/x)³

or, x³ + 1/x³ + 3(x+ 1/x) = (x + 1/x)³

or, x³ + 1/x³ = (x + 1/x)³ - 3(x + 1/x)

or, x³ + 1/x³ = (x + 1/x){ (x + 1/x)² - 3 }

or, x³ + 1/x³ = ±3{(±3)² - 3} . . . . . . . . . {Because x + 1/x = ±3}

or, x³ + 1/x³ = ±3{9 - 3}

or, x³ + 1/x³ = ±3 × 6

So, x³ + 1/x³ = ±18