Problem 52 - College Algebra

Question

Rationalize the denominator. $$\frac{5}{\sqrt{3}-1}$$

Step-by-Step Solution

Answer

The rationized form of \( \frac{5}{\sqrt{3}-1} \) is \( \frac{5\sqrt{3}}{2} + \frac{5}{2} \).

Step 1: Multiply by the conjugate over itself

Multiply the given fraction by the conjugate of denominator over itself, \[\frac{5}{\sqrt{3}-1} * \frac{\sqrt{3}+1}{\sqrt{3}+1}\]

Step 2: Perform the multiplication

First multiply the numerators together,\[5 * (\sqrt{3} + 1) = 5\sqrt{3} + 5\]Then multiply the denominators together, using the expansion formula \( (a+b) * (a-b) = a^{2} - b^{2} \),\[(\sqrt{3}+1) * (\sqrt{3}-1) = (\sqrt{3})^{2} - (1)^{2} = 3 - 1 = 2\]

Step 3: Simplify the fraction

Divide the elements of the numerator by the denominator, \[\frac{5\sqrt{3}+5}{2} = \frac{5\sqrt{3}}{2} + \frac{5}{2}\]