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If \[\dfrac{{5 + 2\sqrt 3 }}{{7 + 4\sqrt 3 }} = a + b\sqrt 3 \], then the value of a and b is ?

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Last updated date: 16th May 2024
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Answer
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Hint: In the given fractional expression, we have an irrational number in the denominator. To simplify the fraction, first we rationalise the denominator in the fraction so as to eliminate radicals from the denominator. To get rid of the irrational number in the denominator, we multiply the numerator and denominator by the conjugate of the denominator. Then by simplification and comparing both sides of the equation, we get the values of a and b.

Complete step-by-step answer:
In the given question, we are provided with an expression \[\left( {\dfrac{{5 + 2\sqrt 3 }}{{7 + 4\sqrt 3 }}} \right)\]. The given expression is in the form of $ \dfrac{a}{b} $ , where a is the numerator and b is the denominator. This form is known as fraction.
We have to simplify the given fractional expression \[\left( {\dfrac{{5 + 2\sqrt 3 }}{{7 + 4\sqrt 3 }}} \right)\] by removing the irrational numbers and surds from the denominator using the method of rationalisation of the denominator.
Generally in mathematics, we don’t desire to have an irrational number in the denominator. So, in order to get rid of the square root, we multiply the fraction by another fraction whose numerator and denominator are equal so as to keep the given fraction unchanged.
Considering the fraction given to us, \[\left( {\dfrac{{5 + 2\sqrt 3 }}{{7 + 4\sqrt 3 }}} \right)\]. We multiply the numerator and denominator of the fraction by the conjugate of the denominator of the fraction. So, the denominator of the fraction \[\left( {\dfrac{{5 + 2\sqrt 3 }}{{7 + 4\sqrt 3 }}} \right)\] is \[\left( {7 + 4\sqrt 3 } \right)\]. Hence, we multiply both numerator and denominator by the conjugate of \[\left( {7 + 4\sqrt 3 } \right)\], that is \[\left( {7 - 4\sqrt 3 } \right)\] so as to get rid of the irrational number in the denominator.
Hence, we get,
\[ \Rightarrow \left( {\dfrac{{5 + 2\sqrt 3 }}{{7 + 4\sqrt 3 }}} \right) \times \left( {\dfrac{{7 - 4\sqrt 3 }}{{7 - 4\sqrt 3 }}} \right)\]
\[ \Rightarrow \dfrac{{\left( {5 + 2\sqrt 3 } \right)\left( {7 - 4\sqrt 3 } \right)}}{{\left( {7 + 4\sqrt 3 } \right)\left( {7 - 4\sqrt 3 } \right)}}\]
Evaluating the denominator of the fraction using the algebraic identity $ \left( {a - b} \right)\left( {a + b} \right) = {a^2} - {b^2} $ , we get,
\[ \Rightarrow \dfrac{{35 + 14\sqrt 3 - 20\sqrt 3 - 8{{\left( {\sqrt 3 } \right)}^2}}}{{{{\left( 7 \right)}^2} - {{\left( {4\sqrt 3 } \right)}^2}}}\]
Simplifying the expression, we get,
\[ \Rightarrow \dfrac{{35 - 6\sqrt 3 - 8\left( 3 \right)}}{{49 - 48}}\]
Adding up the like terms, we get,
\[ \Rightarrow \dfrac{{35 - 6\sqrt 3 - 24}}{1}\]
\[ \Rightarrow 11 - 6\sqrt 3 \]
So, we get the simplified form of the rational expression \[\left( {\dfrac{{5 + 2\sqrt 3 }}{{7 + 4\sqrt 3 }}} \right)\] given to us as \[\left( {11 - 6\sqrt 3 } \right)\] by rationalising the denominator.
Now, we are given in the question that \[\left( {\dfrac{{5 + 2\sqrt 3 }}{{7 + 4\sqrt 3 }}} \right)\] is equal to \[a + b\sqrt 3 \], where the denominator of the rational expression is simplified and we have to find the value of a and b.
So, \[\left( {\dfrac{{5 + 2\sqrt 3 }}{{7 + 4\sqrt 3 }}} \right) = \left( {11 - 6\sqrt 3 } \right) = a + b\sqrt 3 \]
Comparing the sides of equation, we get,
 $ a = 11 $ and $ b = - 6 $
Hence, the value of a is \[11\] and the value of b is \[\left( { - 6} \right)\]
So, the correct answer is “a = 11 and b = -6”.

Note: If the denominator of a fraction has square root, we rationalise the rational number. We multiply the numerator and denominator by the same number so as not to change the value of the fraction and get rid of the irrational number in the denominator. One must take care while doing the calculations so as to be sure of the final answer.