Question

please Rationalise the denominator

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Answer 1

Answer is 1 . . . . . . . .

Answer 2

Solution!!

$\sf \dfrac{\sqrt{5}-2}{\sqrt{5}+2}-\dfrac{\sqrt{5}+2}{\sqrt{5}-2}$

Taking the LCM

$\sf \dfrac{(\sqrt{5}-2)(\sqrt{5}-2)-(\sqrt{5}+2)(\sqrt{5}+2)}{(\sqrt{5}+2)(\sqrt{5}-2)}$

Rationalising the denominator using (a + b)(a - b) = a² - b²

$\sf \dfrac{(\sqrt{5}-2)(\sqrt{5}-2)-(\sqrt{5}+2)(\sqrt{5}+2)}{(\sqrt{5})^{2}-(2)^{2}}$

$\sf \dfrac{(\sqrt{5}-2)(\sqrt{5}-2)-(\sqrt{5}+2)(\sqrt{5}+2)}{5-4}$

$\sf \dfrac{(\sqrt{5}-2)(\sqrt{5}-2)-(\sqrt{5}+2)(\sqrt{5}+2)}{1}$

Write the repeated multiplication in exponential form

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

Expand the expression using (a - b)² = a² + b² - 2ab

(√5)² + (2)² - 2(√5)(2) - (√5 + 2)²

5 + 4 - 4√5 - (√5 + 2)²

9 - 4√5 - (√5 + 2)²

Expand the expression using (a + b)² = a² + b² + 2ab

9 - 4√5 - ((√5)² + (2)² + 2(√5)(2))

9 - 4√5 - (5 + 4 + 4√5)

9 - 4√5 - (9 + 4√5)

Opening the brackets

9 - 4√5 - 9 - 4√5

Collecting the like terms

9 - 9 - 4√5 - 4√5

0 - 8√5

-8√5