Question

Simplify the following expression and write its value in simpler numbers: \[\dfrac{{5 + \sqrt 3 }}{{7 - 4\sqrt 3 }} - \dfrac{{5 - \sqrt 3 }}{{7 + 4\sqrt 3 }}\]

Answer 1

Hint: Recall the definition for rational and irrational numbers. Use rationalization to convert the denominator of each term into a rational number and then evaluate the expression.

Complete step by step solution:

We, now, rationalise \[\dfrac{{5 + \sqrt 3 }}{{7 - 4\sqrt 3 }}\] by multiplying both the numerator and the denominator by \[7 + 4\sqrt 3 \]. Hence, we have as follows:
$\Rightarrow$ \[\dfrac{{5 + \sqrt 3 }}{{7 - 4\sqrt 3 }} = \dfrac{{5 + \sqrt 3 }}{{7 - 4\sqrt 3 }} \times \dfrac{{7 + 4\sqrt 3 }}{{7 + 4\sqrt 3 }}\]
Simplifying using \[(a + b)(a - b) = {a^2} + {b^2}\], we get:
$\Rightarrow$ \[\dfrac{{5 + \sqrt 3 }}{{7 - 4\sqrt 3 }} = \dfrac{{35 + 7\sqrt 3 + 20\sqrt 3 + 12}}{{{7^2} - {{\left( {4\sqrt 3 } \right)}^2}}}\]
$\Rightarrow$ \[\dfrac{{5 + \sqrt 3 }}{{7 - 4\sqrt 3 }} = \dfrac{{47 + 27\sqrt 3 }}{{49 - 48}}\]
$\Rightarrow$ \[\dfrac{{5 + \sqrt 3 }}{{7 - 4\sqrt 3 }} = 47 + 27\sqrt 3 .........(1)\]
We, now, rationalise \[\dfrac{{5 - \sqrt 3 }}{{7 + 4\sqrt 3 }}\] by multiplying both the numerator and the denominator by \[7 - 4\sqrt 3 \]. Hence, we have as follows:
$\Rightarrow$ \[\dfrac{{5 - \sqrt 3 }}{{7 + 4\sqrt 3 }} = \dfrac{{5 - \sqrt 3 }}{{7 + 4\sqrt 3 }} \times \dfrac{{7 - 4\sqrt 3 }}{{7 - 4\sqrt 3 }}\]
Simplifying using \[(a + b)(a - b) = {a^2} + {b^2}\], we get:
$\Rightarrow$ \[\dfrac{{5 - \sqrt 3 }}{{7 + 4\sqrt 3 }} = \dfrac{{35 - 7\sqrt 3 - 20\sqrt 3 + 12}}{{{7^2} - {{\left( {4\sqrt 3 } \right)}^2}}}\]
$\Rightarrow$ \[\dfrac{{5 - \sqrt 3 }}{{7 + 4\sqrt 3 }} = \dfrac{{47 - 27\sqrt 3 }}{{49 - 48}}\]
$\Rightarrow$ \[\dfrac{{5 - \sqrt 3 }}{{7 + 4\sqrt 3 }} = 47 - 27\sqrt 3 .........(2)\]
Using equations (1) and (2), we have:
$\Rightarrow$ \[\dfrac{{5 + \sqrt 3 }}{{7 - 4\sqrt 3 }} - \dfrac{{5 - \sqrt 3 }}{{7 + 4\sqrt 3 }} = 47 + 27\sqrt 3 - (47 - 27\sqrt 3 )\]
$\Rightarrow$ \[\dfrac{{5 + \sqrt 3 }}{{7 - 4\sqrt 3 }} - \dfrac{{5 - \sqrt 3 }}{{7 + 4\sqrt 3 }} = 47 + 27\sqrt 3 - 47 + 27\sqrt 3 \]
$\Rightarrow$ \[\dfrac{{5 + \sqrt 3 }}{{7 - 4\sqrt 3 }} - \dfrac{{5 - \sqrt 3 }}{{7 + 4\sqrt 3 }} = 54\sqrt 3 \]
Hence, the answer is \[54\sqrt 3 \].

Note: When you substitute the rationalized terms do not forget the negative sign in between the terms which might end up with answer 94, which is wrong.