Question

Write the conjugate of the binomial surd $xy\sqrt{z}+yz\sqrt{x}$

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 $(xy\sqrt{z}+yz\sqrt{x})$.

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

$(xy\sqrt{z}+yz\sqrt{x})(xy\sqrt{z}-yz\sqrt{x})={\left(xy\sqrt{z}\right)}^2-{\left(yz\sqrt{x}\right)}^2=z{x}^2{y}^2-x{z}^2{y}^2(\because a^2-b^2=(a+b\left)\right(a-b\left)\right)$

Since $z{x}^2{y}^2-x{z}^2{y}^2$ is a rational number.

Hence, $(xy\sqrt{z}-yz\sqrt{x})$ is the conjugate of $(xy\sqrt{z}+yz\sqrt{x})$.