Question

If $\sqrt {\dfrac{{{\text{6 + 2}}\sqrt 3 }}{{33{\text{ - 19}}\sqrt 3 }}} {\text{ = a + b}}\sqrt 3 $, then a + b =
A. 6
B. 8
C. 10
D. 12

Answer 1

Hint: To solve this problem, we will first square the whole expression given in the question, then we will rationalise the term to solve the given problem. Now, first we will make the left - hand side of the form of right – hand side. After it, we will compare the expressions on both sides to find the value of a and b.

Complete step-by-step solution -
Now, first we will square both sides,
On squaring, we get
$\dfrac{{6{\text{ + 2}}\sqrt 3 }}{{{\text{33 - 19}}\sqrt 3 }}{\text{ = (a + b}}\sqrt 3 {{\text{)}}^2}$
Now, we will rationalise the left – hand side term. Now, rationalisation is done to make the given term simplified. It is done by multiplying and dividing the given term by the term obtained on reversing the sign of the coefficient of under-root in the denominator.
So, on rationalising
$\dfrac{{6{\text{ + 2}}\sqrt 3 }}{{{\text{33 - 19}}\sqrt 3 }}{\text{ }} \times {\text{ }}\dfrac{{33{\text{ + 19}}\sqrt 3 }}{{{\text{33 + 19}}\sqrt 3 }}{\text{ = (a + b}}\sqrt 3 {{\text{)}}^2}$
Here the denominator is $33{\text{ - 19}}\sqrt 3 $ . We reverse the sign of coefficient of under – root, so the term is $33{\text{ + 19}}\sqrt 3 $.
Now, on further solving
$\dfrac{{6({\text{33 + 19}}\sqrt 3 ){\text{ + 2}}\sqrt 3 ({\text{33 + 19}}\sqrt 3 )}}{{({\text{33 - 19}}\sqrt 3 )(33{\text{ + 19}}\sqrt 3 )}}{\text{ = (a + b}}\sqrt 3 {)^2}$
Here in the denominator, we use property $({\text{x + y)(x - y) = }}{{\text{x}}^2}{\text{ - }}{{\text{y}}^2}$
$\dfrac{{312{\text{ + 180}}\sqrt 3 }}{6}{\text{ = (a + b}}\sqrt 3 {)^2}$
$52{\text{ + 30}}\sqrt 3 {\text{ = (a}}{\text{ + b}}\sqrt 3 {)^2}$
Taking under – root both sides, we get
$\sqrt {52{\text{ + 30}}\sqrt 3 {\text{ }}} {\text{ = a + b}}\sqrt 3 $
$\sqrt {52{\text{ + 2}}\sqrt {3 \times {\text{15}} \times {\text{15}}} {\text{ }}} {\text{ = a + b}}\sqrt 3 $
$\sqrt {{\text{27 + 25 + 2}}\sqrt {3 \times 3 \times 3 \times 5 \times 5} } {\text{ = a + b}}\sqrt 3 $
Now, we can see that in the left – hand side, there is an expansion of ${(\sqrt {27} {\text{ + }}\sqrt {25} )^2}$
Therefore, $\sqrt {{{(\sqrt {27} {\text{ + }}\sqrt {25} )}^2}} {\text{ }} = {\text{ (a + b}}\sqrt 3 )$
$\sqrt {27} {\text{ + }}\sqrt {25} {\text{ }} = {\text{ (a + b}}\sqrt 3 )$
${\text{5 + 3}}\sqrt 3 {\text{ = a + b}}\sqrt 3 $
Therefore, on comparing we get a = 5 and b = 3
So, a + b = 5 + 3 = 8
So, option (B) is correct.

Note: When we come up with such types of questions, we will start the question by removing the under – root from the question. For this, we will square the whole expression given in the question. After it, we will rationalise the given term to make it simplified. After simplification, we will take the under – root on the expression and find the values of the required variables and solve the given problem.