Question

Find the values of a and b if 5+2√3/7+4√3=a+b√3

Answer 1

Answer:

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Step-by-step explanation:

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Answer 2

Given:

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

To Find:

Values of a and b

Solution:

We know that,

$\longrightarrow\bf{(x+y)(x-y)=x^{2}-y^{2}}$

It is given that,

$\longrightarrow\mathrm{\dfrac{5+2\sqrt{3}}{7+4\sqrt{3}}=a+b\sqrt{3}}$

On multiplying and dividing 7-4√3 in L.H.S., we get

$\longrightarrow\mathrm{\dfrac{5+2\sqrt{3}}{7+4\sqrt{3}}\times\dfrac{7-4\sqrt{3}}{7-4\sqrt{3}} =a+b\sqrt{3}}$

$\longrightarrow\mathrm{\dfrac{(5+2\sqrt{3})(7-4\sqrt{3})}{(7)^{2}-(4\sqrt{3})^{2}} =a+b\sqrt{3}}$

$\longrightarrow\sf{\dfrac{5(7-4\sqrt{3})+2\sqrt{3}(7-4\sqrt{3})}{49-48} =a+b\sqrt{3}}$

$\longrightarrow\sf{\dfrac{35-20\sqrt{3}+14\sqrt{3}-24}{1} =a+b\sqrt{3}}$

$\longrightarrow\sf{35-24-20\sqrt{3}+14\sqrt{3}=a+b\sqrt{3}}$

$\longrightarrow\sf{11-6\sqrt{3} =a+b\sqrt{3}}$

On comparing the constant term and coefficient of √3 of both the sides, we get

$\longrightarrow\mathrm{\pink{a=11}}$

$\longrightarrow\mathrm{\green{b=-6}}$

Hence, the value of a is 11 and bis -6.