Problem 51 - College Algebra

Question

Rationalize the denominator. $$\frac{7}{\sqrt{5}-2}$$

Step-by-Step Solution

Answer

\(7\sqrt{5} + 14 \)

Step 1: Identify the Conjugate

The conjugate of a binomial term \(\sqrt{5}-2\) is the term with the same numbers but with a different sign between those numbers. Therefore, the conjugate is \(\sqrt{5}+2\). We will use this conjugate to rationalize the denominator.

Step 2: Multiply by Conjugate

Next step is to multiply both the numerator and the denominator by the conjugate \(\sqrt{5}+2\). This is equivalent to multiplying the fraction by 1 and therefore does not change its value.\[\frac{7}{\sqrt{5}-2}\] multiplied by \[\frac{\sqrt{5}+2}{\sqrt{5}+2}\] gives: \[\frac{7(\sqrt{5}+2)}{(\sqrt{5}-2)(\sqrt{5}+2)}\].

Step 3: Simplify

Now, simplify the numerator and denominator separately. The numerator is straightforward multiplication, and the denominator uses the difference of squares formula (a^2 - b^2 = (a+b)(a-b)). This yields: \[ \frac{7\sqrt{5}+14}{5-4} \].

Step 4: Final Simplification

Further simplify to get the final answer: \[ \frac{7\sqrt{5}+14}{1} = 7\sqrt{5} + 14 \].