Question

F x4 + 1/x4 = 47, then find the value of x3 + 1/x3

Answer 1

x⁴+1/x⁴= 47
(x²)²+(1/x²)²+2 = 49
we know that.
a²+b²+2ab = (a+b)²

(x²+1/x²)² = 49
x²+1/x² = √49 = 7

(x)²+(1/x)²+2 = 9
(x+1/x)² = 9
(x+1/x) = √9 = 3

(x+1/x)³ = x³+1/x³+3(x)(1/x) [ x+1/x ]
(3)³ = x³+1/x³+3(3)
27 = x³+1/x³+9
27-9 = 18 => x³+1/x³


hope this helps
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Answer 2

$x^4+ \frac{1}{x^4} = 47\\x^4+\frac{1}{x^4} + 2 = 49\\(x^2 + \frac{1}{x^2})^2 = (7)^2\\ x^2+\frac{1}{x^2} + 2 = 7+2\\(x^2 + \frac{1}{x})^2 = (3)^2\\x+\frac{1}{x} = 3\\(x+\frac{1}{x})^3 = x^3 + \frac{1}{x^3} + 3(x+\frac{1}{x})\\27 = x^3 + \frac{1}{x^3} + 9\\x^3 + \frac{1}{x^3} = 18$