Question

Write the conjugates of the binomial surds $x+3\sqrt{y}$

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 $(x+3\sqrt{y})$.

When we multiply this with the difference of the same two terms, that is, with $(x-3\sqrt{y})$, the product is:

$(x+3\sqrt{y})(x-3\sqrt{y})={\left(x\right)}^2-{\left(3\sqrt{y}\right)}^2=x^2-9y(\because a^2-b^2=(a+b\left)\right(a-b\left)\right)$

Since $x^2-9y$ is a rational number.

Hence, $(x-3\sqrt{y})$ is the conjugate of $(x+3\sqrt{y})$.