Question

Simplify each of the following by rationalizing the denominator. (sqrt(5) - 2)/(sqrt(5) + 2) - (sqrt(5) + 2)/(sqrt(5) - 2)​

Answer 1

$\large\sf\underline{Given\::}$

• $\sf\:\frac{\sqrt{5}-2}{\sqrt{5}+2}-\frac{\sqrt{5}+2}{\sqrt{5}-2}$

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$\large\sf\underline{To\::}$

• Rationalise the given expression.

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$\large\sf\underline{Things\:to\:know\::}$

Rationalization means the process of getting rid of any surds in the denominator. Simplifying a fraction with surd simply refers to rationalization.

So now new question arises :

$\small\bf\:How\:do\:we\:get\:rid\:of\:the\:surds\:from\:the\:denominator\:?$

This is pretty simple we just need to multiply the fraction by the conjugate of its denominator.

Now :

$\small\bf\:What\:is\:meant\:by\:conjugate\:?$

Let's understand conjugate with the help of an example. If we are given x + y and said to find it's conjugate then, it's conjugate would be x - y.

So conjugate is an expression formed by changing the sign of the given expression.

Hope I sound clear now :D

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$\large\sf\underline{Solution\::}$

$\sf\:\frac{\sqrt{5}-2}{\sqrt{5}+2}-\frac{\sqrt{5}+2}{\sqrt{5}-2}$

• Taking LCM of (√5 + 2) and (√5 - 2) = (√5 + 2) (√5 - 2)

$\sf\implies\:\frac{(\sqrt{5}-2)(\sqrt{5}-2)-(\sqrt{5}+2)(\sqrt{5}+2)}{(\sqrt{5}+2)(\sqrt{5}-2)}$

$\sf\implies\:\frac{(\sqrt{5}-2)^{2}-(\sqrt{5}+2)^{2}}{(\sqrt{5}+2)(\sqrt{5}-2)}$

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Using some identities :

In numerator :

• $\sf\:(a+b)^{2} = a^{2} + 2ab + b^{2}$

• $\sf\:(a-b)^{2} = a^{2} - 2ab + b^{2}$

In denominator :

• $\sf\:a^{2}-b^{2}=(a+b)(a-b)$

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$\sf\implies\:\frac{[(\sqrt{5})^{2}-2 \times \sqrt{5} \times 2 + (2)^{2}]-[(\sqrt{5})^{2}+ 2 \times \sqrt{5} \times 2 + (2)^{2}]}{(\sqrt{5})^{2}-(2)^{2}}$

$\sf\implies\:\frac{[5-4\sqrt{5} +4]-[5+ 4 \sqrt{5} + 4]}{5-4}$

• Opening the brackets

$\sf\implies\:\frac{5-4\sqrt{5} +4-5- 4 \sqrt{5} - 4}{1}$

$\sf\implies\:5-4\sqrt{5} +4-5- 4 \sqrt{5} - 4$

• Arranging the like terms

$\sf\implies\:5+4-5-4-4\sqrt{5} -4\sqrt{5}$

• Proceeding with simple calculations

$\sf\implies\:9-5-4-4\sqrt{5} -4\sqrt{5}$

$\sf\implies\:4-4-4\sqrt{5} -4\sqrt{5}$

$\sf\implies\:0-4\sqrt{5} -4\sqrt{5}$

$\sf\implies\:-4\sqrt{5} -4\sqrt{5}$

$\small{\underline{\boxed{\mathrm\red{\implies\:-8\sqrt{5}}}}}$ ★

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$\dag\:\underline{\sf So\:the\:simplified\:value\:is\:(-8\sqrt{5})}$ .

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Note :- We could have simplified the given expression separately i.e., first $\sf\:\frac{\sqrt{5}-2}{\sqrt{5}+2}$ and then $\frac{\sqrt{5}+2}{\sqrt{5}-2}$ . But here I have simplified them together. Either way we will get the same result that is (-8√5).

Thank you :D

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‎!! Hope it helps !!