Question

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

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

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

$(\frac{1}{2}+\sqrt{2})(\frac{1}{2}-\sqrt{2})={\left(\frac{1}{2}\right)}^2-{\left(\sqrt{2}\right)}^2=\frac{1}{4}-2=\frac{1-8}{4}=-\frac{7}{4}(\because a^2-b^2=(a+b\left)\right(a-b\left)\right)$

Since $-\frac{7}{4}$ is a rational number.

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