Question

question
express as a fraction in simplest form with a rational denominator:
$\frac{4}{2 - sqrt{15}}$

Explanation:

Step1: Rationalize denominator with conjugate

Multiply numerator and denominator by $2+\sqrt{15}$:
$$\frac{4}{2-\sqrt{15}} \times \frac{2+\sqrt{15}}{2+\sqrt{15}} = \frac{4(2+\sqrt{15})}{(2-\sqrt{15})(2+\sqrt{15})}$$

Step2: Expand denominator using difference of squares

Apply $(a-b)(a+b)=a^2-b^2$:
$$\frac{4(2+\sqrt{15})}{2^2 - (\sqrt{15})^2} = \frac{8+4\sqrt{15}}{4-15}$$

Step3: Simplify denominator

Calculate the denominator value:
$$\frac{8+4\sqrt{15}}{-11}$$

Step4: Rewrite in standard form

Move negative sign to numerator:
$$-\frac{8+4\sqrt{15}}{11}$$

Answer:

$-\frac{8+4\sqrt{15}}{11}$