Question

if a=1-√2, find the value of (a-1/a)3​

Answer 1

Answer:

Required numeric value of ( a - 1 / a )^3 is 8.

Step-by-step explanation:

Given,

a = 1 - √2

$\implies \dfrac{1}{a}=\dfrac{1}{1-\sqrt2}$

Using Rationalisation : Multiply and divide by the original denominator with opposite signs between the rational and irrational number so that the result will be rational number.

So, here, we have to multiply and divide by 1 + √2.

$\implies \dfrac{1}{a}=\dfrac{1}{1-\sqrt2}\times\dfrac{1+\sqrt2}{1+\sqrt2}\\\\\\\implies\dfrac{1}{a}=\dfrac{1+\sqrt2}{(1-\sqrt2)(1+\sqrt2)}$

From the properties of expansion :

• ( a + b )( a - b ) = a^2 - b^2

$\implies \dfrac{1}{a}=\dfrac{1+\sqrt2}{(1)^2-(\sqrt2)^2}\\\\\\\implies\dfrac{1}{a}=\dfrac{1+\sqrt2}{1-2}\\\\\\\implies\dfrac{1}{a}=-1-\sqrt2$

Therefore,

= > ( a - 1 / a )^3

= > { ( 1 - √2 ) - ( - 1 - √2 ) }^3

= > { 1 - √2 + 1 + √2 }^3

= > ( 2 )^3

= > 8

Hence the required numeric value of ( a - 1 / a )^3 is 8.

Answer 2

Step-by-step explanation:

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