Question 85
What is the value of a - b if it is known that $$\sqrt{\ \dfrac{7\ +\ 4\sqrt{\ 3}}{3}}\ =\ a\ +\ b\sqrt{\ 3}$$? (where a and b are rational numbers)
Solution
We are given that,
$$\sqrt{\ \dfrac{7\ +\ 4\sqrt{\ 3}}{3}}\ =\ a\ +\ b\sqrt{\ 3}$$
LHS can be written as,
$$\sqrt{\ \dfrac{\left(7\ +\ 4\sqrt{\ 3}\right)\times\ 3}{3\ \times\ 3}}\ =\ \sqrt{\dfrac{21\ +\ 12\sqrt{\ 3}}{9}\ }=\ \ \sqrt{\dfrac{3^2\ +\ 2\times\ 3\times\ 2\sqrt{\ 3}\ +\ \left(2\sqrt{\ 3}\right)^2}{3^2}\ }\ =\ \sqrt{\left(\ \dfrac{3\ +\ 2\sqrt{\ 3}}{3}\right)^{^2}}\ =\ 1\ +\ \dfrac{2}{3}\sqrt{\ 3}$$
The value of a is 1, and b is $$\dfrac{2}{3}$$.
The value of $$a\ -\ b\ =\ 1\ -\ \dfrac{2}{3}\ =\ \dfrac{1}{3}$$
The correct answer is option C.
