Question

Rationalise the denominator

Answer 1

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Heya



(6 - 4√3)/(6 + 4√3)

$= \frac{6 - 4 \sqrt{3} }{6 + 4 \sqrt{3} } \times \frac{6 - 4 \sqrt{3} }{6 - 4 \sqrt{3} } \\ \\ = \frac{ ({6 - 4 \sqrt{3} )}^{2} }{(6 + 4 \sqrt{3})(6 - 4 \sqrt{3}) } \\ \\ = \frac{ {(6)}^{2} + {(4 \sqrt{3}) }^{2} - 2(6)(4 \sqrt{3}) }{ {(6)}^{2} - {(4 \sqrt{3}) }^{2} } \\ \\ = \frac{36 + 48 - 48 \sqrt{3} }{36 - 48} \\ \\ = \frac{84 - 48 \sqrt{3} }{ - 12} \\ \\ = \frac{ - (84 - 48 \sqrt{3}) }{12} \\ \\ = \frac{48 \sqrt{3} - 84}{12} \\ \\ = \frac{12(4 \sqrt{3} - 7) }{12} \\ \\ = 4 \sqrt{3} - 7$



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Identities used :-

(a - b)^2 = a^2 + b^2 - 2ab

(a + b)(a - b) = a^2 - b^2



Hope this helps you.

Answer 2

HEYA. MATE

YOUR ANSWER

$\frac{6 \: - \: 4 \sqrt{3} }{6 + 4 \sqrt{3} } = \\ \\ = \geqslant \frac{6 - \sqrt[4]{3} }{6 + \sqrt[4]{3} } \: \: \times \: \frac{6 - \sqrt[4]{3} }{6 - \sqrt[4]{3} } \\ \\ = \geqslant \frac{(6 - \sqrt[4]{3) {}^{2} } }{(6) {}^{2} \: - ( \sqrt[4]{3 }) {}^{2} } \\ \\ = \geqslant \frac{ {(6)}^{2} \: + ( \sqrt[4]{3} ) {}^{2} - 2 \times 6 \times \sqrt[4]{3} }{36 - 48} \\ \\ = \geqslant \frac{36 + 48 - 48 \sqrt{3} }{ - 12} \\ \\ = \geqslant \frac{36 \sqrt{3} }{ - 12} \\ \\ = \geqslant - 3 \sqrt{3} answer$