Question

if x⁴+1/x⁴=47. then find the value of x³+1/x³?​

Answer 1

Answer:

∴ The value of x³ + 1/x³ is 18.

Step-by-step explanation:

Given:-

x⁴ + 1/x⁴ = 47

This can be written as:

⟹ (x²)² + (1/x²)² = 47

We know that,

a² + b² = (a + b)² - 2ab

So,

⟹ (x² + 1/x²)² - 2 * (x²) * (1/x²) = 47

⟹ (x² + 1/x²)² - 2 = 47

⟹ (x² + 1/x²)² = 47 + 2

⟹ (x² + 1/x²)² = 49

⟹ x² + 1/x² = √49

⟹ x² + 1/x² = 7

Again using a² + b² = (a + b)² - 2ab we get,

⟹ (x + 1/x)² - 2 (x)(1/x) = 7

⟹ (x + 1/x)² - 2 = 7

⟹ (x + 1/x)² = 7 + 2

⟹ (x + 1/x)² = 9

⟹ (x + 1/x) = √9

⟹ (x + 1/x) = 3 -- equation (1).

Now,

Cubing both sides we get,

⟹ (x + 1/x)³ = (3)³

using (a + b)³ = a³ + b³ + 3ab(a + b) we get,

⟹ x³ + 1/x³ + 3 (x) (1/x) (x + 1/x) = 27

Substituting x + 1/x = ± 3 [ from equation (1) ] in LHS we get,

⟹ x³ + 1/x³ + 3(3) = 27

⟹ x³ + 1/x³ + 9 = 27

⟹ x³ + 1/x³ = 27 - 9

⟹ x³ + 1/x³ = 18

∴ The value of x³ + 1/x³ is 18.

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