Question

Rationalize the denominator.

$\huge \tt\frac{4}{7 + 4 \sqrt{3}}$

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Answer 1

Answer:

refer to the above attachment

Answer 2

Question

Rationalise the denominator.

$\large{\tt{\dfrac{4}{7 + 4 \sqrt{3}}}}$

Answer :-

$\maltese \: \: \: \: \: \sf{\dfrac{4}{7 + 4 \sqrt{3}} = \red{28 + 16 \sqrt{3}}}$

EXPLANATION

$\longmapsto \: \: \tt{\dfrac{4}{7 + 4 \sqrt{3}}}$

On multiplying (7-4√3)/(7-4√3),

$\longmapsto \: \: \tt{\dfrac{4}{7 + 4 \sqrt{3}} \times \dfrac{7 - 4 \sqrt{3}}{7 - 4 \sqrt{3}}}$

$\longmapsto \: \: \tt{ \dfrac{4(7 - 4 \sqrt{3})}{{(7)}^{2} - {(4 \sqrt{3})}^{2}}}$

Simplifying the numerator 4(7-4√3),

$\longmapsto \: \: \tt{ \dfrac{4(7) - 4(4 \sqrt{3})}{{(7)}^{2} - {(4 \sqrt{3})}^{2}}}$

$\longmapsto \: \: \tt{ \dfrac{28 - 16 \sqrt{3}}{{(7)}^{2} - {(4 \sqrt{3})}^{2}}}$

On simplifying, the denominator (7)² = 49 and (4√3)² = 16*3 = 48,

$\longmapsto \: \: \tt{ \dfrac{28 - 16 \sqrt{3}}{49 - {(16 \times 3)}}}$

$\longmapsto \: \: \tt{ \dfrac{28 - 16 \sqrt{3}}{49 - {48}}}$

On Subtracting 48 from 49,

$\longmapsto \: \: \tt{ \dfrac{28 - 16 \sqrt{3}}{1}}$

$\longmapsto \: \underline{\boxed{ \bf{ANSWER} = \frak{28 - 16 \sqrt{3}}}} \: \: \: \dag$

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Hence, The required number is 28 - 16√3.