Question

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Answer 1

Answer:

The required value of x - 1/x is -4√2

Step-by-step explanation:

$\sf x=\dfrac{1}{3+2\sqrt{2}}$

First, let's rationalize the denominator.

Rationalizing factor = 3 - 2√2

Multiply and divide the fraction (x) by the rationalizing factor.

 $\implies \sf \dfrac{1}{3+2\sqrt{2}} \times \dfrac{3-2\sqrt{2}}{3-2\sqrt{2}} \\\\ \implies \sf \dfrac{3-2\sqrt{2}}{(3+2\sqrt{2})(3-2\sqrt{2})} \\\\ \implies \sf \dfrac{3-2\sqrt{2}}{3^2-(2\sqrt{2})^2} \ \ [ \because (a+b)(a-b)=a^2-b^2] \\\\ \implies \sf \dfrac{3-2\sqrt{2}}{9-4(2)} \\\\ \implies \sf \dfrac{3-2\sqrt{2}}{9-8} \\\\ \implies \sf 3-2\sqrt{2}$

Now,

$\sf \dfrac{1}{x}=\dfrac{1}{\dfrac{1}{3+2\sqrt{2}}} \\\\ \implies \sf \dfrac{1}{x}=3+2\sqrt{2}$

$\longmapsto \sf x-\dfrac{1}{x} \\\\ \implies \sf 3-2\sqrt{2}-(3+2\sqrt{2}) \\\\ \implies \sf 3-2\sqrt{2}-3-2\sqrt{2} \\\\ \implies \sf -4\sqrt{2}$

Therefore, the required value of x - 1/x is -4√2

Answer 2

• refer attachment for detailed answers

• rationalize the value of x

• apply the value of in given condition

• equate and evaluate

• -4√2 is answers