Question

Rationalise the denominator and simplify

Answer 1

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Answer 2

Explanation:

`\bf \underline{Given-} \\`

`(5 + 2 √(3) )/(7 + 4 √(3) ) \\`

`\bf \underline{What\: to\: do-} \\`

To rationalise the denominator

`\bf \underline{Solution-} \\`

`\textsf{We have,}`

`(5 + 2 √(3) )/(7 + 4 √(3) ) \\ \\`

`\textsf{The denominator is 5+2√3. Multiplying the numerator and denominator by 7-4√3,}\\`

`\textsf{we get,}\\`

`⟹(5 + 2 √(3) )/(7 + 4 √(3) ) * (7 - 4 √(3) )/(7 - 4 √(3) ) \\ \\`

`⟹ ((5 + 2 √(3) )(7 - 4 √(3)) )/((7 + 4 √(3) )(7 - 4 √(3)) ) \\ \\`

`\textsf{⬤ Applying Algebraic Identity</p><p>(a+b)(a-b) = a² - b² to the denominator}\\`

`\textsf{We get,}\\`

`⟹ \frac{(5 + 2 √(3) )(7 - 4 √(3)) }{(7 {)}^(2) - (4 √(3) {)}^(2) } \\ \\`

`⟹ (35 + 14 √(3) - 20 √(3) - 8 √(3 * 3) )/(49 - 48) \\ \\`

`⟹ (35 + 14 √(3) - 20 √(3) - 24 )/(1) \\ \\`

`⟹(35 - 24) - 6 √(3) \\ \\`

`⟹11 - 6 √(3) \: \: \: \tt \red{ Ans}. \\ \\`

`\bf \underline{Hence \:the \:denominator\: is\: rationalised.}\\`