Question

If x+1/x=2,prove that x2+1/x2=x3+1/x3=x4+1/x4​

Answer 1

Solution

Given:::X+1/X=2

now...

x²+1/x²=(x+1/x)²-2(x)(1/x)

=2²-2

=4-2

=2

x³+1/x³=(x+1/x){x²+1/x²-x.(1/x)}

=2(2-1)

=2

x⁴+1/x⁴=(x²+1/x²)²-2(x²)(1/x²)

=2²-2

=4-2

=2

therefore...x²+1/x²=x³+1/x³=x⁴+1/x⁴ [proved]

Answer 2

Step-by-step explanation:

Given :-

$\\ x + \frac{1}{x} = 2$

To prove:-

$x {}^{2} + \frac{1}{x {}^{2} } = x {}^{3} + \frac{1}{x {}^{3} } = x {}^{4} + \frac{1}{x {}^{4} }$

$\huge\bf\orange{Solution:-}$

$x {}^{2} + \frac{1}{x {}^{2} } = (x + \frac{1}{x} ) {}^{2} - 2 \times x \times \frac{1}{x} \\ = 2 {}^{2} - 2 \\ = 4 - 2 \\ = 2$

Then,

$x {}^{3} + \frac{1}{x {}^{3} } = (x + \frac{1}{x} ) {}^{3} - 3 \times x \times \frac{1}{x} (x + \frac{1}{x} ) \\ = 2 {}</h3><h3>^{3} - 3 \times 2 \\ = 8 - 6 \\ = 2$

Again,

$x {}^{4} + \frac{1}{x {}^{4} } = (x {}^{2} + \frac{1}{x {}^{2} } ) {}^{2} - 2 \times x {}^{2} \times \frac{1}{x {}^{2} } \\ = 2 {}^{2} - 2 \\ = 4 - 2 \\ = 2$

So,

$x {}^{2} + \frac{1}{x {}^{2} } = x {}^{3} + \frac{1}{x {}^{3} } = x {}^{4} + \frac{1}{x {}^{4} }$

[Proved]

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