Question

Write the conjugates of binomial surd given as $\sqrt{a}+\sqrt{b}$

Answer 1

We know that the when sum of two terms and the difference of the same two terms are multiplied, the product is always a rational number.

Let us apply this concept to a binomial surd $(\sqrt{a}+\sqrt{b})$.

When we multiply this with the difference of the same two terms, that is, with $(\sqrt{a}-\sqrt{b})$, the product is:

$(\sqrt{a}+\sqrt{b})(\sqrt{a}-\sqrt{b})={\left(\sqrt{a}\right)}^2-{\left(\sqrt{b}\right)}^2=a-b(\because a^2-b^2=(a+b\left)\right(a-b\left)\right)$

Since $a-b$ is a rational number.

Hence, $(\sqrt{a}-\sqrt{b})$ is the conjugate of $(\sqrt{a}+\sqrt{b})$.