Question

Write the conjugates of the binomial surds given as $\sqrt{x}-2\sqrt{y}$

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 $(\sqrt{x}-2\sqrt{y})$.

When we multiply this with the sum of the same two terms, that is, with $(\sqrt{x}+2\sqrt{y})$, the product is:

$(\sqrt{x}-2\sqrt{y})(\sqrt{x}+2\sqrt{y})={\left(\sqrt{x}\right)}^2-{\left(2\sqrt{y}\right)}^2=x-4y(\because a^2-b^2=(a+b\left)\right(a-b\left)\right)$

Since $x-4y$ is a rational number.

Hence, $(\sqrt{x}+2\sqrt{y})$ is the conjugate of $(\sqrt{x}-2\sqrt{y})$.