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Chapter 0: Problem 51

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

Short Answer

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

Step by step solution

01

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.
02

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: \[\frac{7(\sqrt{5}+2)}{(\sqrt{5}-2)(\sqrt{5}+2)}\].
03

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: \[ \frac{7\sqrt{5}+14}{5-4} \].
04

Final Simplification

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

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Most popular questions from this chapter

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