Simplify:

$$
\left(\frac{1}{5+\sqrt{3}}\right) + \left(\frac{1}{5-\sqrt{3}}\right) - \left(\frac{10}{25-3}\right)
$$

Question

Simplify:

$$
\left(\frac{1}{5+\sqrt{3}}\right) + \left(\frac{1}{5-\sqrt{3}}\right) - \left(\frac{10}{25-3}\right)
$$

Accepted Answer

1. Note the identity: $(5+\sqrt3)(5-\sqrt3)=25-3$.

2. Write the expression as

$\displaystyle \frac{5+\sqrt3}{5-\sqrt3}+\frac{5-\sqrt3}{25-3}-\frac{25-3}{5+\sqrt3}$.

3. Use the identity to simplify: $\dfrac{5-\sqrt3}{25-3}=\dfrac{1}{5+\sqrt3}$ and $\dfrac{25-3}{5+\sqrt3}=5-\sqrt3$.

4. So the whole sum becomes $\dfrac{5+\sqrt3}{5-\sqrt3}+\dfrac{1}{5+\sqrt3}-(5-\sqrt3)$.

5. Combine the last two terms and simplify. They cancel the first term exactly. The result is $0$.

FINAL_ANSWER: a

Answer

Answer

Answer