Question

Write the conjugates of the binomial surd $3\sqrt{p}-2\sqrt{q}$

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 $(3\sqrt{p}-2\sqrt{q})$.

When we multiply this with the sum of the same two terms, that is, with $(3\sqrt{p}+2\sqrt{q})$, the product is:

$(3\sqrt{p}-2\sqrt{q})(3\sqrt{p}+2\sqrt{q})={\left(3\sqrt{p}\right)}^2-{\left(2\sqrt{q}\right)}^2=9p-4q(\because a^2-b^2=(a+b\left)\right(a-b\left)\right)$

Since $9p-4q$ is a rational number.

Hence, $(3\sqrt{p}+2\sqrt{q})$ is the conjugate of $(3\sqrt{p}-2\sqrt{q})$.