Question

Find the rationalizing factor of: $$\left(\sqrt{3}+\sqrt{10}-\sqrt{5}\right)$$
A. $$\left(\sqrt{3}+\sqrt{10}+\sqrt{5}\right)\left(8-2\sqrt{30}\right)$$
B. $$\left(\sqrt{3}+\sqrt{10}+\sqrt{5}\right)$$
C. $$\left(\sqrt{3}+\sqrt{10}+\sqrt{5}\right)\left(8+2\sqrt{30}\right)$$
D. None of the above

Answer 1

Hint: The factor of multiplication by which rationalization is done is called the rationalizing factor. So, rationalize the given expression by using the formula $$\left(a-b\right)\left(a+b\right)=a^2-b^2$$ twice to obtain the required answer.

Complete step-by-step answer:
Rationalising factor means the term by which we convert irrational numbers to rational numbers.
So, it means we have to choose a term by which we make $$\left(\sqrt{3}+\sqrt{10}-\sqrt{5}\right)$$ is a rational number.
We know that, $$\left(a-b\right)\left(a+b\right)=a^2-b^2$$
Since,
 $$\begin{gathered} \left(\sqrt{3}+\sqrt{10}-\sqrt{5}\right)\left(\sqrt{3}+\sqrt{10}+\sqrt{5}\right)={\left(\sqrt{3}+\sqrt{10}\right)}^2-{\left(\sqrt{5}\right)}^2 \\ \left(\sqrt{3}+\sqrt{10}-\sqrt{5}\right)\left(\sqrt{3}+\sqrt{10}+\sqrt{5}\right)=3+10+2\sqrt{3}\sqrt{10}-5 \\ \left(\sqrt{3}+\sqrt{10}-\sqrt{5}\right)\left(\sqrt{3}+\sqrt{10}+\sqrt{5}\right)=8+2\sqrt{30} \end{gathered}$$
And
$$\begin{gathered} \left(8+2\sqrt{30}\right)\left(8-2\sqrt{30}\right)=8^2-{\left(2\sqrt{30}\right)}^2 \\ \left(8+2\sqrt{30}\right)\left(8-2\sqrt{30}\right)=64-4\times 30 \\ \left(8+2\sqrt{30}\right)\left(8-2\sqrt{30}\right)=64-120=-56 \end{gathered}$$
Therefore, $$\left(\sqrt{3}+\sqrt{10}-\sqrt{5}\right)\left(\sqrt{3}+\sqrt{10}+\sqrt{5}\right)\left(8-2\sqrt{30}\right)=-56$$ which is a rational number.
Hence, $$\left(\sqrt{3}+\sqrt{10}+\sqrt{5}\right)\left(8-2\sqrt{30}\right)$$ is a rational number $$\left(\sqrt{3}+\sqrt{10}-\sqrt{5}\right)$$.
Thus, the correct option is A. $$\left(\sqrt{3}+\sqrt{10}+\sqrt{5}\right)\left(8-2\sqrt{30}\right)$$

Note: In this question, the given expression $$\left(\sqrt{3}+\sqrt{10}-\sqrt{5}\right)$$ is a Surd. If the product of two or more surds is a rational number then they are rationalizing factors to each other. Sometimes we divide to get the rationalizing factor.