Question

How do you simplify the following expression: $\dfrac{x}{\sqrt{x+5}-\sqrt{5}}$?

Answer 1

Hint: We can use the identity theorem of ${{a}^{2}}-{{b}^{2}}=\left( a+b \right)\left( a-b \right)$ for the denominator. We multiply $\left( \sqrt{x+5}+\sqrt{5} \right)$ to both denominator and the numerator. We simplify the denominator and eliminate $x$.

Complete step by step solution:
The given surds expression is $\dfrac{x}{\sqrt{x+5}-\sqrt{5}}$. We have to apply the identity of ${{a}^{2}}-{{b}^{2}}=\left( a+b \right)\left( a-b \right)$.
We multiply $\left( \sqrt{x+5}+\sqrt{5} \right)$ to both denominator and the numerator. This is the conjugate form of $\left( \sqrt{x+5}-\sqrt{5} \right)$.
Now the fraction becomes $\dfrac{x}{\sqrt{x+5}-\sqrt{5}}=\dfrac{x\left( \sqrt{x+5}+\sqrt{5} \right)}{\left( \sqrt{x+5}-\sqrt{5} \right)\left( \sqrt{x+5}+\sqrt{5} \right)}$.
We assume the values $a=\sqrt{x+5};b=\sqrt{5}$ for ${{a}^{2}}-{{b}^{2}}=\left( a+b \right)\left( a-b \right)$.
Therefore, $\left( \sqrt{x+5}-\sqrt{5} \right)\left( \sqrt{x+5}+\sqrt{5} \right)={{\left( \sqrt{x+5} \right)}^{2}}-{{\left( \sqrt{5} \right)}^{2}}$.
Simplifying we get $\left( \sqrt{x+5}-\sqrt{5} \right)\left( \sqrt{x+5}+\sqrt{5} \right)={{\left( \sqrt{x+5} \right)}^{2}}-{{\left( \sqrt{5} \right)}^{2}}$
$\left( \sqrt{x+5}-\sqrt{5} \right)\left( \sqrt{x+5}+\sqrt{5} \right)=x+5-5=x$.
The fraction becomes $\dfrac{x}{\sqrt{x+5}-\sqrt{5}}=\dfrac{x\left( \sqrt{x+5}+\sqrt{5} \right)}{\left( \sqrt{x+5}-\sqrt{5} \right)\left( \sqrt{x+5}+\sqrt{5} \right)}=\dfrac{x\left( \sqrt{x+5}+\sqrt{5} \right)}{x}$.
We eliminate the variable $x$ to find $\dfrac{x}{\sqrt{x+5}-\sqrt{5}}=\dfrac{x\left( \sqrt{x+5}+\sqrt{5} \right)}{x}=\left( \sqrt{x+5}+\sqrt{5} \right)$.
Therefore, the simplified form of $\dfrac{x}{\sqrt{x+5}-\sqrt{5}}$ is $\left( \sqrt{x+5}+\sqrt{5} \right)$.

Note:
Instead of multiplying $\left( \sqrt{x+5}+\sqrt{5} \right)$, we can also form the variable $x$ as $x=x+5-5$. We try to form the identity of ${{a}^{2}}-{{b}^{2}}=\left( a+b \right)\left( a-b \right)$ where we take $x=\left( x+5 \right)-\left( 5 \right)={{\left( \sqrt{x+5} \right)}^{2}}-{{\left( \sqrt{5} \right)}^{2}}$.
Now we break it to get ${{\left( \sqrt{x+5} \right)}^{2}}-{{\left( \sqrt{5} \right)}^{2}}=\left( \sqrt{x+5}-\sqrt{5} \right)\left( \sqrt{x+5}+\sqrt{5} \right)$.
Then we can eliminate the part of $\left( \sqrt{x+5}-\sqrt{5} \right)$ to simplify.