Question

Multiply the surd $\left( \sqrt{5}+\sqrt{2} \right)$ by _____ to get a rational number.

Answer 1

Hint: We will be using the concept of surds and indices to solve the problem. Also, we will be using the concepts of irrational numbers while simplifying the solution. We will also be using the algebraic identities like,
$\left( a+b \right)\left( a-b \right)={{a}^{2}}-{{b}^{2}}$

Complete step-by-step solution -
Now, we have been given a surd $\left( \sqrt{5}+\sqrt{2} \right)$. Now, we know that to rationalize a surd. We have to multiply it by its math conjugate.
Now, math conjugates a number formed by changing the sign between the two terms in binomial. For instance we have the conjugate of x + y is x – y. We can also say that x + y is a conjugate of x – y.
Now, the special thing about a surd and its conjugate is that if we multiply both of them we get a rational number. For example: if $\sqrt{a}+\sqrt{b}$ is a surd and $\sqrt{a}-\sqrt{b}$ is its conjugate. So, $\begin{align}
  & \left( \sqrt{a}+\sqrt{b} \right)\left( \sqrt{a}-\sqrt{b} \right)={{\left( \sqrt{a} \right)}^{2}}-{{\left( \sqrt{b} \right)}^{2}} \\
 & =a-b \\
\end{align}$
So, making $\left( \sqrt{a}+\sqrt{b} \right)$ rational. Hence, to convert $\left( \sqrt{5}+\sqrt{2} \right)$ into a rational number we have to multiply it by its conjugate that is $\sqrt{5}-\sqrt{2}$.
So, we have,
$\left( \sqrt{5}+\sqrt{2} \right)\left( \sqrt{5}-\sqrt{2} \right)$
Now, we know the algebraic identity that $\left( a+b \right)\left( a-b \right)={{a}^{2}}-{{b}^{2}}$. Therefore,
$\begin{align}
  & \left( \sqrt{5}+\sqrt{2} \right)\left( \sqrt{5}-\sqrt{2} \right)={{\left( \sqrt{5} \right)}^{2}}-{{\left( \sqrt{2} \right)}^{2}} \\
 & =5-2 \\
 & =3 \\
\end{align}$
Which is a rational number. Therefore, multiply $\left( \sqrt{5}+\sqrt{2} \right)$ by $\left( \sqrt{5}-\sqrt{2} \right)$ to get a rational number.

Note: To solve these type of questions it is important to remember that,
$\left( a+b \right)\left( a-b \right)={{a}^{2}}-{{b}^{2}}$
Also, we can remember that if $\left( \sqrt{a}+\sqrt{b} \right)$ is a surd then it can be rationalized by $\sqrt{a}-\sqrt{b}$ and vice - versa. Here we need to be aware about the difference between rational and irrational numbers.