Question

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



please help​

Answer 1

Step-by-step explanation:

Given :-

x+(1/x) = 2

To find:-

Prove that x²+1/x² = x³+1/x³ = x⁴+1/x⁴ .

Solution:-

Given that

x+(1/x) = 2 -----------(1)

On squaring both sides then

[x+(1/x)]² = 2²

=> x²+(1/x)²+2(x)(1/x) = 4

Since (a+b)² = a²+2ab+b²

Where ,a = x and b = 1/x

=> x²+(1/x²)+2 = 4

=> x²+(1/x²) = 4-2

=> x²+(1/x²) = 2-------(2)

We know that

(a+b)³ = a³+b³+3ab(a+b)

=> a³+b³ = (a+b)³-3ab(a+b)

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

=> x³+(1/x³) = 2³-3(2) (from (1))

=> x³+(1/x³) = 8-6

=> x³+(1/x³) = 2---------(3)

On squaring both sides of the equation (2)

[x²+(1/x²)]² = 2²

=> (x²)²+(1/x²)²+2(x²)(1/x²) = 4

Since (a+b)² = a²+2ab+b²

Where ,a = x² and b = 1/x²

=> x⁴+(1/x⁴)+2 = 4

=> x⁴+(1/x⁴) = 4-2

=> x⁴+(1/x⁴) = 2--------(4)

From (2),(3)&(4) then

x²+1/x² = x³+1/x³ = x⁴+1/x⁴

Answer :-

If x+1/x = 2 then x²+1/x² = x³+1/x³ = x⁴+1/x⁴

Used formulae:-

• (a+b)² = a²+2ab+b²
• (a+b)³ = a³+b³+3ab(a+b)
• a³+b³ = (a+b)³-3ab(a+b)
• (a^m)^n = a^(mn)