Question

Find the rationalising factor of the given binomial surd $\sqrt{1+y}-\sqrt{1-y}$

Answer 1

The given binomial surd $\sqrt{1+y}-\sqrt{1-y}$ can be rewritten as $(1+y{)}^{\frac{1}{2}}-(1-y{)}^{\frac{1}{2}}$.

Let $a=(1+y{)}^{\frac{1}{2}}$ and $b=(1-y{)}^{\frac{1}{2}}$, therefore,

$a^2={\left(\right(1+y{)}^{\frac{1}{2}})}^2\Rightarrow a^2=(1+y{)}^{\frac{1}{2}\times 2}\Rightarrow a^2=1+y$

Also,

$b^2={\left(\right(1-y{)}^{\frac{1}{2}})}^2\Rightarrow b^2=(1-y{)}^{\frac{1}{2}\times 2}\Rightarrow b^2=1-y$

Thus, $a^2-b^2=(1+y)-(1-y)=1-1+y+y=2y$

But we also know that the formula for $a^2-b^2=(a+b)(a-b)$, therefore,

$a^2-b^2=(a+b)(a-b)\Rightarrow 2y=(\sqrt{1+y}+\sqrt{1-y})(\sqrt{1+y}-\sqrt{1-y})$

Since $2y$ is a rational number.

Hence, $(\sqrt{1+y}+\sqrt{1-y})$ is the rationalising factor of $(\sqrt{1+y}-\sqrt{1-y})$.