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Mathematics

If x = 13+22\dfrac{1}{3 + 2\sqrt2} , then find the value of x−1xx - \dfrac{1}{x}.

Rational Irrational Nos

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Answer

Given,

x=13+22….(i)x =\dfrac{1}{3 + 2\sqrt{2}} \qquad \text{….(i)} \\[1.5em]

Let us rationalise the denominator,

x=13+22×3−223−22=3−2232−(22)2=3−229−8=(3−22)∴x=(3−22)x = \dfrac{1}{3 + 2\sqrt{2}} ×\dfrac{3 - 2\sqrt{2}}{3 - 2\sqrt{2}} \\[1.5em] = \dfrac{3 - 2\sqrt{2}}{3^2 - (2\sqrt{2})^2} \\[1.5em] = \dfrac{3 - 2\sqrt{2}}{9 - 8} \\[1.5em] = (3 - 2\sqrt{2}) \\[1.5em] \therefore x = (3 - 2\sqrt{2})

From (i) we get,

1x=3+22∴x−1x=(3−22)−(3+22)=3−22−3−22⇒x−1x=−42\dfrac{1}{x} = 3 + 2\sqrt{2} \\[1.5em] \therefore x - \dfrac{1}{x} = (3 - 2\sqrt{2}) - (3 + 2\sqrt{2}) = 3 - 2\sqrt{2} - 3 - 2\sqrt{2}\\[1.5em] \Rightarrow\bold{ x - \dfrac{1}{x} = -4\sqrt{2}} \\[1.5em]

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