Question

Prove a^2+1/a^2=a^3+1/a^3=a^4+1/a^4of a+1/a=2

Answer 1

$\text{a}^{2}+ \frac{1}{\text{a}^{2} } = \text{a}^{3} +\frac{1}{\text{a}^3} = \text{a}^4+\frac{1}{\text{a}^4} = 2$ is proved as the a = 1

Step-by-step explanation:

Given :

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

To Prove :

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

Proof :

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

a² + 1 = 2a

a²-2a + 1 = 0

a²-2ab +b² = (a-b)²

∴ a²-2a + 1 = (a - 1 )²

(a - 1 )² = 0

Taking square root on both sides we get,

√(a-1)² = 0

a- 1 = 0

∴ a = 1

Substituting a = 1 in the given equation

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

1 + 1 = 1+ 1=1+ 1= 2

2 = 2 = 2 =2

Thus proved.

To Learn More.....

1. Verify the identity (a-b)²=a²-2ab+b² geometrically by taking a=3 , b=2

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