Question

5+2root3 = a+ broot3
7+4root3
​

Answer 1

Appropriate Question :-

$\rm :\longmapsto\:\dfrac{5 + 2 \sqrt{3} }{7 + 4 \sqrt{3} } = a + b \sqrt{3}, \: find \: a \: and \: b$

Basic Concept Used :-

Method of Rationalization :-

Rationalization is the process of removing radicals from denominator.

Multiple and Divide the given expression by conjugate of denominator.

$\huge\underline{\bf{Solution-}}$

Given that

$\rm :\longmapsto\:\dfrac{5 + 2 \sqrt{3} }{7 + 4 \sqrt{3} } = a + b \sqrt{3} \:$

$\rm :\longmapsto\:\dfrac{5 + 2 \sqrt{3} }{7 + 4 \sqrt{3}} \times \dfrac{7 - 4 \sqrt{3} }{7 - 4 \sqrt{3} } = a + b \sqrt{3} \:$

$\rm :\longmapsto\:\dfrac{35 - 20 \sqrt{3} + 14 \sqrt{3} - 24 }{ {7}^{2} - {(4 \sqrt{3})}^{2} } = a + b \sqrt{3}, \:$

$\rm :\longmapsto\:\dfrac{11 - 6 \sqrt{3} }{49 - 48} = a + b \sqrt{3}$

$\rm :\longmapsto\:11 - 6 \sqrt{3} = a + b \sqrt{3}$

On comparing, we get

$\: \: \: \: \: \: \: \: \: \: \: \: \purple{\boxed{ \bf \: a = 11}} \: \: \: and \: \: \: \purple{\boxed{ \bf \: b = - 6}}$

Answer 2

Step-by-step explanation:

$\bf{ \underline{Given-}}$

$\frac{5 + 2 \sqrt{3} }{7 + 4 \sqrt{3} } = \bf \: a + b \sqrt{3}$

$\bf \underline{To \: find \: out-}$

$\rm{Value \: of \: a \: and \: b \: in \: given \: expression.}$

$\bf \underline{Solution-}$

Given expression

$\frac{5 + 2 \sqrt{3} }{7 + 4 \sqrt{3} } = \bf \: a + b \sqrt{3}$

The denominator is 7 + 4√3.

We know that

Rationalising factor of a + b√c = a - b√c.

So, the rationalising factor of 7 +4√3 = 7-4√3.

On rationalising the denominator them

$\longrightarrow \: \frac{5 + 2 \sqrt{3} }{7 + 4 \sqrt{3} } \times \frac{7 - 4 \sqrt{3} }{7 - 4 \sqrt{3} }$

$\longrightarrow \: \frac{(5 + 2 \sqrt{3})(7 - 4 \sqrt{3}) }{(7 + 4 \sqrt{3})(7 - 4 \sqrt{3}) }$

Now, applying algebraic identity in denominator because it is in the form of;

(a+b)(a-b) = a² - b²

Where, we have to put in our expression: a = 7 and b = 4√3 , we get

$\longrightarrow \: \frac{(5 + 2 \sqrt{3} )(7 - 4 \sqrt{3}) }{(7 {)}^{2} - (4 \sqrt{3} {)}^{2} }$

$\longrightarrow \: \frac{(5 + 2 \sqrt{3} )(7 - 4 \sqrt{3}) }{49 - 48}$

Subtract 49 from 48 in denominator to get 1.

$\longrightarrow \: \frac{(5 + 2 \sqrt{3} )(7 - 4 \sqrt{3}) }{1}$

$\longrightarrow \: (5 + 2 \sqrt{3} )(7 - 4 \sqrt{3} )$

Now, multiply both term left side to right side.

$\longrightarrow \: 35 + 14 \sqrt{3} - 20 \sqrt{3} - 8 \sqrt{3 \times 3 }$

$\longrightarrow \: 35 + 14 \sqrt{3} - 20 \sqrt{3} - 24$

$\longrightarrow \: (35 - 24) - 6 \sqrt{3}$

$\longrightarrow \: 11 - 6 \sqrt{3}$

$\bf\therefore \: 11 - 6 \sqrt{3} = a + b \sqrt{3}$

On, comparing with R.H.S , we have

a = 11 and b = -6

$\bf \underline{Hence, value \: of \: a = 11 \: and \: b = - 6.}$

Used Formulae:

(a+b)(a-b) = a² - b²

Rationalising factor of a + b√c = a - b√c.