Question

Simplify the expression. $$\frac{9}{5-\sqrt{7}}$$

Short answer

The simplified form of the given expression is \(\frac{5 + \sqrt{7}}{2}\).

Step-by-step solution

Identify the Conjugate

The conjugate of \(5-\sqrt{7}\) is \(5+\sqrt{7}\). The conjugate of a binomial \(a - b\) is \(a + b\). We change the sign in the middle to get the conjugate.

Multiply by the Conjugate

Next, multiply both the numerator and the denominator by the conjugate, which is \(5+\sqrt{7}\). This gives: \[ \frac{9}{5-\sqrt{7}} \times \frac{5+\sqrt{7}}{5+\sqrt{7}} \] Remember to multiply the expression out which means both in numerator and denominator we follow the rule (a+b)(a-b)= \(a^2 - b^2\).

Simplify the Resulting Expression

The above expression simplifies to: \[ \frac{9*(5+\sqrt{7})}{5^2 - (\sqrt{7})^2} = \frac{45 + 9\sqrt{7}}{25 - 7} \] Simplify the denominator further to get: \[ \frac{45 + 9\sqrt{7}}{18} \]

Simplify Further

The final simplified expression will be: \[ \frac{5 + \sqrt{7}}{2} \]. Here both terms in the numerator are divisible by 9.