Question

Simplify the following: $\dfrac{2}{\sqrt{5}+\sqrt{3}}+\dfrac{1}{\sqrt{3}+\sqrt{2}}-\dfrac{3}{\sqrt{5}+\sqrt{2}}$

Answer 1

Hint: To simplify the given expression, we need to rationalize the denominators and simplify. To rationalize, we need to multiply the numerator and denominator by the conjugate of the denominator of each term and simplify.

Complete step-by-step answer:
The given question requires us to simplify the expression $\dfrac{2}{\sqrt{5}+\sqrt{3}}+\dfrac{1}{\sqrt{3}+\sqrt{2}}-\dfrac{3}{\sqrt{5}+\sqrt{2}}.$ For this, we need to rationalize the terms and we do this by multiplying the conjugate of the denominator with both the numerator and denominator of the term. For the first term, the conjugate of the denominator $\sqrt{5}+\sqrt{3}$ is $\sqrt{5}-\sqrt{3}.$ Similarly for the second term the conjugate of $\sqrt{3}+\sqrt{2}$ is $\sqrt{3}+\sqrt{2}.$ For the third term, the conjugate of $\sqrt{5}+\sqrt{2}$ is $\sqrt{5}-\sqrt{2}.$
Multiplying these with their respective numerators and denominators,
$\Rightarrow \dfrac{2\left( \sqrt{5}-\sqrt{3} \right)}{\left( \sqrt{5}+\sqrt{3} \right)\left( \sqrt{5}-\sqrt{3} \right)}+\dfrac{1\left( \sqrt{3}-\sqrt{2} \right)}{\left( \sqrt{3}+\sqrt{2} \right)\left( \sqrt{3}-\sqrt{2} \right)}-\dfrac{3\left( \sqrt{5}-\sqrt{2} \right)}{\left( \sqrt{5}+\sqrt{2} \right)\left( \sqrt{5}-\sqrt{2} \right)}$
To simplify the denominators, we use the expansion $\left( a+b \right).\left( a-b \right)={{a}^{2}}-{{b}^{2}}.$ Using this formula,
$\Rightarrow \dfrac{2\left( \sqrt{5}-\sqrt{3} \right)}{{{\left( \sqrt{5} \right)}^{2}}-{{\left( \sqrt{3} \right)}^{2}}}+\dfrac{1\left( \sqrt{3}-\sqrt{2} \right)}{{{\left( \sqrt{3} \right)}^{2}}-{{\left( \sqrt{2} \right)}^{2}}}-\dfrac{3\left( \sqrt{5}-\sqrt{2} \right)}{{{\left( \sqrt{5} \right)}^{2}}-{{\left( \sqrt{2} \right)}^{2}}}$
Square and root cancel each other and we get,
$\Rightarrow \dfrac{2\left( \sqrt{5}-\sqrt{3} \right)}{5-3}+\dfrac{1\left( \sqrt{3}-\sqrt{2} \right)}{3-2}-\dfrac{3\left( \sqrt{5}-\sqrt{2} \right)}{5-2}$
Subtracting the terms in the denominator,
$\Rightarrow \dfrac{2\left( \sqrt{5}-\sqrt{3} \right)}{2}+\dfrac{1\left( \sqrt{3}-\sqrt{2} \right)}{1}-\dfrac{3\left( \sqrt{5}-\sqrt{2} \right)}{3}$
Since we have common factors in the numerator and denominator, we cancel them and obtain the following,
$\Rightarrow \left( \sqrt{5}-\sqrt{3} \right)+\left( \sqrt{3}-\sqrt{2} \right)-\left( \sqrt{5}-\sqrt{2} \right)$
Taking the terms outside the brackets by multiplying with the negative sign for the last term,
$\Rightarrow \sqrt{5}-\sqrt{3}+\sqrt{3}-\sqrt{2}-\sqrt{5}+\sqrt{2}$
We subtract the corresponding terms and we can see that every term gets cancelled out.
$\Rightarrow 0$
Hence, the simplified value of $\dfrac{2}{\sqrt{5}+\sqrt{3}}+\dfrac{1}{\sqrt{3}+\sqrt{2}}-\dfrac{3}{\sqrt{5}+\sqrt{2}}$ is 0.

Note: Students need to have a good knowledge in the topics rationalization to solve this question. Care must be taken while rationalizing and apply the formula appropriately, and we need to carefully take the conjugate of the denominator only. Taking the conjugate of the numerator only makes the problem more complex. We can solve this question directly if we know the values of $\sqrt{5},\sqrt{3}$ and $\sqrt{2}.$ But this method can be a little more complex since it involves the operations on decimals.