Question

Rationalise the denominator:-
$4 \div \sqrt{6} - \sqrt{4}$
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Answer 1

Answer:

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$\sf \dashrightarrow \ \dfrac{2\sqrt{6} + 4}{1}$

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Step-by-step explanation:

We're asked to rationalise the following:

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$\sf \dashrightarrow \ \dfrac{4}{\sqrt{6} - \sqrt{4}}$

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• In order to rationalise the denominator, we'll have to multiply both the numerator and the denominator of the fraction by the conjugate of the denominator.

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• Conjugating an expression with two terms basically changes the sign between the two terms from + (plus) to - (minus) or vice-versa.

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For example;

• Conjugate of 5 + 3 is 5 - 3

• Conjugate of 5 - √3 is 5 + √3

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We have;

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$\sf \dashrightarrow \ \dfrac{4}{\sqrt{6} - \sqrt{4}}$

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In order to rationalise the denominator, we'll have to multiply the numerator and denominator by the conjugate of the denominator.

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The conjugate of the denominator is √6 + √4, on multiplying it with the numerator and denominator we get;

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$\sf \dashrightarrow \ \dfrac{4}{\sqrt{6} - \sqrt{4}} \ \times \ \dfrac{\sqrt{6} + \sqrt{4}}{\sqrt{6} + \sqrt{4}}$

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$\sf \dashrightarrow \ \dfrac{4\Big\{\sqrt{6} + \sqrt{4}\Big\}}{\Big\{\sqrt{6} - \sqrt{4}\Big\}\Big\{\sqrt{6} + \sqrt{4}\Big\}}$

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On applying the identity (a - b)(a + b) = a² - b² we get;

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$\sf \dashrightarrow \ \dfrac{4\Big\{\sqrt{6} + \sqrt{4}\Big\}}{\Big\{\sqrt{6}\Big\}^{2} - \Big\{\sqrt{4}\Big\}^{2}}$

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$\sf \dashrightarrow \ \dfrac{4\Big\{\sqrt{6} + \sqrt{4}\Big\}}{6 - 4}$

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$\sf \dashrightarrow \ \dfrac{4\Big\{\sqrt{6} + \sqrt{4}\Big\}}{2}$

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On dividing 4 and 2 we get;

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$\sf \dashrightarrow \ \dfrac{2\Big\{\sqrt{6} + \sqrt{4}\Big\}}{1}$

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$\sf \dashrightarrow \ \dfrac{2\sqrt{6} + 2\sqrt{4}}{1}$

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$\sf \dashrightarrow \ \dfrac{2\sqrt{6} + 2\sqrt{2 \times 2}}{1}$

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$\sf \dashrightarrow \ \dfrac{2\sqrt{6} + (2 \times 2)}{1}$

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$\sf \dashrightarrow \ \dfrac{2\sqrt{6} + 4}{1}$

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Hence rationalised.

Answer 2

Answer:

Question :-

Rationalise the denominator:-

$4 \div \sqrt{6} - \sqrt{4}$

Required Answer :-

$\frac{ 4\sqrt{6} + 4 \sqrt{4} }{2}$

Note :-

Plz refer to the attachment