Question

x⁴+1/x⁴=623, find x³+1/x³​

Answer 1

EXPLANATION.

⇒ x⁴ + 1/x⁴ = 623.

As we know that,

⇒ (x² + 1/x²)² = x⁴ + 1/x⁴ + 2(x²)(1/x²).

⇒ (x² + 1/x²)² = x⁴ + 1/x⁴ + 2.

Put the value of x⁴ + 1/x⁴ = 623 in the equation, we get.

⇒ (x² + 1/x²)² = 623 + 2.

⇒ (x² + 1/x²)² = 625.

⇒ (x² + 1/x²) = √625.

⇒ (x² + 1/x²) = 25.

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 values of x² + 1/x² = 25 in the equation, we get.

⇒ (x + 1/x)² = 25 + 2.

⇒ (x + 1/x)² = 27.

⇒ (x + 1/x) = √27.

⇒ (x + 1/x) = 3√3.

Cubing on both sides of the equation, we get.

⇒ (x + 1/x)³ = (3√3)³.

⇒ (x)³ + 3(x)²(1/x) + 3(x)(1/x)² + (1/x)³ = (81√3).

⇒ x³ + 3x + 3/x + 1/x³ = 81√3.

⇒ x³ + 3(x + 1/x) + 1/x³ = 81√3.

Put the values of x + 1/x = 3√3 in the equation, we get.

⇒ x³ + 3(3√3) + 1/x³ = 81√3.

⇒ x³ + 1/x³ + 9√3 = 81√3.

⇒ x³ + 1/x³ = 81√3 - 9√3.

⇒ x³ + 1/x³ = 72√3.

Answer 2

Step-by-step explanation:

x⁴+1/x⁴=623

Adding 2 on both sides

x⁴+1/x⁴+2=623+2

(x²+1/x²)²=625

x²+1/x²=√625=25

Adding 2 on both sides

x²+1/x²+2=25+2

(x+1/x)²=27

x+1/x=√27=3√3

Cube on both sides

(x+1/x)³=x³+1/x³+3(x+1/x)

(3√3)³=x³+1/x³+3×3√3

81√3=x³+1/x³+9√3

x³+1/x³=81√3-9√3=72√3