Question

Write the conjugates of the binomial surd $\frac{1}{2}x+\frac{1}{2}\sqrt{y}$

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 $(\frac{1}{2}x+\frac{1}{2}\sqrt{y})$.

When we multiply this with the difference of the same two terms, that is, with $(\frac{1}{2}x-\frac{1}{2}\sqrt{y})$, the product is:

$(\frac{1}{2}x+\frac{1}{2}\sqrt{y})(\frac{1}{2}x-\frac{1}{2}\sqrt{y})={\left(\frac{1}{2}x\right)}^2-{\left(\frac{1}{2}\sqrt{y}\right)}^2=\frac{1}{4}{x}^2-\frac{1}{4}y=\frac{1}{4}(x^2-y)(\because a^2-b^2=(a+b\left)\right(a-b\left)\right)$

Since $\frac{1}{4}(x^2-y)$ is a rational number.

Hence, $(\frac{1}{2}x-\frac{1}{2}\sqrt{y})$ is the conjugate of $(\frac{1}{2}x+\frac{1}{2}\sqrt{y})$.