Question

If both a and b are rational numbers, find the value of a and b from the following equation:-

(2+3√5)/(4+5√5)=a+b√5

​

Answer 1

Given:-

• $\mathsf{\;\dfrac{2 + 3\sqrt{5}}{4 + 5\sqrt{5}}}$

Need To Find Out :-

• Value of a and b

Solution :-

$\mathsf\red {{:\implies\;\dfrac{2 + 3\sqrt{5}}{4 + 5\sqrt{5}}}}$

$\mathsf{Multiplying\;numerator\;and\;denominator\;with\;4 - 5\sqrt{5},\;we\;get :}$

$\mathsf{:\implies \dfrac{(2 + 3\sqrt{5})(4 - 5\sqrt{5})}{(4 + 5\sqrt{5})(4 - 5\sqrt{5})}}$

★  We know that :-

• $\mathsf\purple{{ (a + b)(a - b) = a² - b²}}$

$\mathsf{:\implies \dfrac{(2)(4) - (2)(5\sqrt{5}) + (4)(3\sqrt{5}) - (5\sqrt{5})(3\sqrt{5})}{(4)^2 - (5\sqrt{5})^2}}$

$\mathsf\ {\: \: :\implies \dfrac{8 - 10\sqrt{5} + 12\sqrt{5} - (15)(\sqrt{5})^2}{16 - 25(\sqrt{5})^2}}$

$\mathsf{\: \: \: \: \: \: \: \: \: \: \: \: :\implies \dfrac{8 + 2\sqrt{5} - (15)(5)}{16 - 25(5)}}$

$\mathsf {\: \: \: \: \: \: \: \: \: \: \: \: :\implies \dfrac{8 + 2\sqrt{5} - 75}{16 - 125}}$

$\mathsf{\: \:\: \: \: \:\:\: \: \: \: \: \: \: \: \: :\implies \dfrac{2\sqrt{5} - 67}{-109}}$

$\mathsf{\: \: \: \: \:\:\: \: \: \: \: \: \: \: \: \: :\implies \dfrac{67}{109} - \dfrac{2\sqrt{5}}{109}}$

$\sf\purple{{\: \: \: \: \: \:\: \: \:\: \: \: \: \: \: \: \: :\implies \dfrac{67}{109} - \dfrac{2\sqrt{5}}{109} = a + b\sqrt{5}}}$

★ Comparing on both sides, We get :-

•  $\mathsf\purple{{a = \dfrac{67}{109}}}$

•  $\mathsf\purple{{b = \dfrac{-2}{109}}}$

Answer 2

Given : Both a and b are rational numbers : $\sf \dfrac{ 2+ 3\sqrt{5} }{4 + 5\sqrt {5}} = a + b \sqrt {5}$

Exigency To Find : The value of a and b .

⠀⠀⠀⠀⠀━━━━━━━━━━━━━━━━━━━⠀

Given that ,

$\qquad \dag\:\:\bigg\lgroup \sf{ \dfrac{ 2+ 3\sqrt{5} }{4 + 5\sqrt {5}} = a + b \sqrt {5} }\bigg\rgroup$

⠀⠀⠀⠀⠀Here ,

$\qquad :\implies \bf L.H.S \:\: = \:\: \sf \dfrac{ 2+ 3\sqrt{5} }{4 + 5\sqrt {5}}$

$\qquad :\implies \bf R.H.S \:\: = \:\: \sf a+b\sqrt {5}$

⠀⠀⠀⠀⠀⠀$\underline {\boldsymbol{\star\:Now \: By \: Solving \: the \: L.H.S \::}}$

$\qquad :\implies \bf L.H.S \:\: = \:\: \sf \dfrac{ 2+ 3\sqrt{5} }{4 + 5\sqrt {5}}$

$\qquad :\implies \:\: \sf \dfrac{ 2+ 3\sqrt{5} }{4 + 5\sqrt {5}}$

$\:\: \dag \qquad \sf Now , \: By \: \; \bf Rationalizing \: \sf the \: denominator \::$

$\qquad :\implies \:\: \sf \dfrac{ 2+ 3\sqrt{5} }{4 + 5\sqrt {5}}$

$\qquad :\implies \:\: \sf \dfrac{ (2+ 3\sqrt{5} )\times ( 4 - 5\sqrt {5} )}{(4 + 5\sqrt {5}) \times ( 4 - 5\sqrt{5}) }$

$\dag\:\:\sf{ As,\:We\:know\:that\::}$

$\qquad \dag\:\:\bigg\lgroup \sf{Algebraic \:Indentity \: \:: ( a + b) (a - b) = a^2 - b^2 }\bigg\rgroup$

⠀⠀⠀⠀⠀⠀$\underline {\boldsymbol{\star\:Now \: By \: Applying \: this \::}}$

$\qquad :\implies \:\: \sf \dfrac{ (2+ 3\sqrt{5} )\times ( 4 - 5\sqrt {5} )}{(4 + 5\sqrt {5}) \times ( 4 - 5\sqrt{5}) }$

$\qquad :\implies \:\: \sf \dfrac{ (2+ 3\sqrt{5} )\times ( 4 - 5\sqrt {5} )}{(4)^2 - (5\sqrt {5})^2 }$

$\qquad :\implies \:\: \sf \dfrac{ (2+ 3\sqrt{5} )\times ( 4 - 5\sqrt {5} )}{16 - (5\sqrt {5})^2 }$

$\qquad :\implies \:\: \sf \dfrac{ (2+ 3\sqrt{5} )\times ( 4 - 5\sqrt {5} )}{16 - 25(5) }$

$\qquad :\implies \:\: \sf \dfrac{ (2+ 3\sqrt{5} )\times ( 4 - 5\sqrt {5} )}{16 - 125 }$

$\qquad :\implies \:\: \sf \dfrac{ 2 ( 4 - 5\sqrt {5} ) + 3 \sqrt{5} (4 - 5\sqrt {5}) }{16 - 125 }$

$\qquad :\implies \:\: \sf \dfrac{ 8 - 2(5\sqrt {5} ) + 3 \sqrt{5} (4 - 5\sqrt {5}) }{16 - 125 }$

$\qquad :\implies \:\: \sf \dfrac{ 8 - 10\sqrt {5} + 12 \sqrt{5} - 15(\sqrt {5})^2 }{16 - 125 }$

$\qquad :\implies \:\: \sf \dfrac{ 8 - 10\sqrt {5} + 12 \sqrt{5} - 15(5) }{16 - 125 }$

$\qquad :\implies \:\: \sf \dfrac{ 8 - 10\sqrt {5} + 12 \sqrt{5} - 75 }{16 - 125 }$

$\qquad :\implies \:\: \sf \dfrac{ 8 + 2 \sqrt{5} - 75 }{16 - 125 }$

$\qquad :\implies \:\: \sf \dfrac{ 8 + 2 \sqrt{5} - 75 }{-109 }$

$\qquad :\implies \:\: \sf \dfrac{ -67+ 2 \sqrt{5} }{-109 }$

$\qquad :\implies \:\: \sf \dfrac{ -67}{-109} + \dfrac{ 2 \sqrt{5} }{-109 }$

$\qquad :\implies \:\: \sf \dfrac{ \cancel {-} 67}{\cancel {-}109} + \dfrac{ 2 \sqrt{5} }{-109 }$

[ ( -ve ) negative sign will eliminated from both numerator and Denominator]

$\qquad :\implies \:\: \sf \dfrac{ 67}{109} + \dfrac{ 2 \sqrt{5} }{-109 }$

[ Now , By Shifting ( -ve ) negative sign to numerator . ]

$\qquad :\implies \:\: \sf \dfrac{ 67}{109} - \dfrac{ 2 \sqrt{5} }{109 }$

$\qquad :\implies \:\bf \: L.H.S \:: \sf \dfrac{ 67}{109} - \dfrac{ 2 \sqrt{5} }{109 }$

⠀⠀⠀⠀⠀⠀$\underline {\boldsymbol{\star\:Now \: By \: Comparing \: the \: L.H.S \: to\: R.H.S \::}}$

$\qquad :\implies \:\bf \: L.H.S \:: \sf \dfrac{ 67}{109} - \dfrac{ 2 \sqrt{5} }{109 }$

$\qquad :\implies \bf R.H.S \:\: = \:\: \sf a+b\sqrt {5}$

⠀⠀⠀⠀⠀⠀We get ,

• $\bf a \: = \: \dfrac{ 67}{109} \:$
• $\bf b \: = \: \dfrac{ -2}{109} \:$

Therefore,

⠀⠀⠀⠀⠀$\therefore {\underline{ \sf \:The\:Value \:of\:a \:and \:b \:are\:\bf \dfrac{ 67}{109}\: \& \:\dfrac{ -2}{109}\:\:\sf \:\: , \:respectively \:. }}$

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