Question

Write the conjugates of the binomial surd $\sqrt{8}-5$

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 $(\sqrt{8}-5)$.

When we multiply this with the sum of the same two terms, that is, with $(\sqrt{8}+5)$, the product is:

$(\sqrt{8}-5)(\sqrt{8}+5)={\left(\sqrt{8}\right)}^2-{\left(5\right)}^2=8-25=-17(\because a^2-b^2=(a+b\left)\right(a-b\left)\right)$

Since $-17$ is a rational number.

Hence, $(\sqrt{8}+5)$ is the conjugate of $(\sqrt{8}-5)$.