Question

Write the conjugates of the binomial surd $10\sqrt{2}+3\sqrt{5}$

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 $(10\sqrt{2}+3\sqrt{5})$.

When we multiply this with the difference of the same two terms, that is, with $(10\sqrt{2}-3\sqrt{5})$, the product is:

$(10\sqrt{2}+3\sqrt{5})(10\sqrt{2}-3\sqrt{5})={\left(10\sqrt{2}\right)}^2-{\left(3\sqrt{5}\right)}^2=(10\times 2)-(3\times 5)=20-15=5(\because a^2-b^2=(a+b\left)\right(a-b\left)\right)$

Since $5$ is a rational number.

Hence, $(10\sqrt{2}-3\sqrt{5})$ is the conjugate of $(10\sqrt{2}+3\sqrt{5})$.