Consider $\left(\sqrt{5}-\sqrt{3}\right)\left(\sqrt{5}+\sqrt{3}\right)$. Multiplication can be transformed into difference of squares using the rule: $\left(a-b\right)\left(a+b\right)=a^{2}-b^{2}$.
Factor $8=2^{2}\times 2$. Rewrite the square root of the product $\sqrt{2^{2}\times 2}$ as the product of square roots $\sqrt{2^{2}}\sqrt{2}$. Take the square root of $2^{2}$.
To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $2$ and $4$ is $4$. Multiply $\frac{5\left(\sqrt{5}+\sqrt{3}\right)}{2}$ times $\frac{2}{2}$.
Since $\frac{2\times 5\left(\sqrt{5}+\sqrt{3}\right)}{4}$ and $\frac{3\sqrt{2}}{4}$ have the same denominator, subtract them by subtracting their numerators.
