Question

Write the conjugate of the binomial surd $x\sqrt{a}+y\sqrt{b}$

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 $(x\sqrt{a}+y\sqrt{b})$.

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

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

Since $a{x}^2-b{y}^2$ is a rational number.

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