Question

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Answer 1

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(6 - 4√2) / (6 + 4√2)

• By rationalizing the denominator :-

= (6 - 4√2)(6 - 4√2) / (6 + 4√2)(6 - 4√2)

= (6 - 4√2)² / [ (6)² - (4√2)² ]

= (6² + 4√2² - 2 × 6 × 4√2) / (36 - 32)

= (36 + 32 - 48√2) / 4

= (68 - 48√2) / 4

= [ 4(17 - 12√2) ] / 4

• 4 cancels in the numerator and the denominator :-

= 17 - 12√2

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Answer 2

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

• Multiply both numerator & denominator by " 6 - 4√2 "

$⟹ \frac{6 - 4 \sqrt{2} }{6 + 4\sqrt{2} } \times \frac{6 - 4\sqrt{2} }{6 - 4\sqrt{2} }$

$⟹ \frac{( {6 - 4 \sqrt{2}) }^{2} }{(6 +4 \sqrt{2})(6 - 4\sqrt{2} )}$

• [ (a - b)² = a² + b² - 2ab ]

• [ (a + b)(a - b) = a² - b² ]

$⟹ \frac{ {(6)}^{2} + {( 4\sqrt{2}) }^{2} - 2(6)( 4\sqrt{2} ) }{ {(6)}^{2} - {(4 \sqrt{2} )}^{2} }$

$⟹ \frac{36 + 32 - 12 \sqrt{4} }{36 - 32}$

$⟹ \frac{68 - 48\sqrt{2} }{4}$

$⟹ \frac{68}{4} - \frac{48 \sqrt{4} }{4}$

$⟹17 - 12 \sqrt{2}$

$\underline{\boxed{\bf{\purple{∴ \frac{6 - 4 \sqrt{2} }{6 + 4\sqrt{2} } = 17 - 12 \sqrt{2} }}}}$

Step-by-step explanation:

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