x⁴+1/x⁴=623, find x³+1/x³
Answers
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.
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