Problem 1 - Algebra 2 and Trigonometry

Question

Justin simplified \(\frac{7}{2 \sqrt{7}}\) by first writing 7 as \(\sqrt{49}\) and then dividing numerator and denominator by \(\sqrt{7}\). a. Show that Justin's solution is correct. b. Can \(\frac{7}{2 \sqrt{5}}\) be simplified by using the same procedure? Explain why or why not.

Step-by-Step Solution

Answer

Yes, Justin's simplification is correct. No, \(\frac{7}{2\sqrt{5}}\) can't be simplified the same way as \(\frac{\sqrt{49}}{\sqrt{5}}\) isn't a perfect square.

Step 1: Simplify the numerator

First, rewrite the numerator 7 as the square root: \(7 = \sqrt{49}\). Hence, the expression becomes \(\frac{\sqrt{49}}{2\sqrt{7}}\).

Step 2: Simplify by dividing by \(\sqrt{7}\)

Divide both the numerator and the denominator by \(\sqrt{7}\): \(\frac{\sqrt{49}}{2\sqrt{7}} = \frac{\sqrt{49}/\sqrt{7}}{2}\).

Step 3: Simplify the square root division

Recall that \(\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}\); hence this simplifies to \(\frac{\sqrt{49/7}}{2} = \frac{\sqrt{7}}{2}\).

Step 4: Conclusion for part a

Since \(\frac{7}{2\sqrt{7}} = \frac{\sqrt{7}}{2}\), Justin’s simplification is indeed correct.

Step 5: Attempt simplification for \(\frac{7}{2\sqrt{5}}\)

Rewrite 7 as \(\sqrt{49}\) and try a similar simplification: \(\frac{\sqrt{49}}{2\sqrt{5}}\).

Step 6: Try dividing by \(\sqrt{5}\)

If we divide the numerator and the denominator by \(\sqrt{5}\), we get \(\frac{\sqrt{49}/\sqrt{5}}{2}\).

Step 7: Evaluate \(\frac{\sqrt{49}}{\sqrt{5}}\)

Since \(\frac{\sqrt{49}}{\sqrt{5}} = \sqrt{\frac{49}{5}}\) and \(\sqrt{\frac{49}{5}}\) does not simplify nicely (as it’s not a perfect square), the original expression doesn’t simplify further using this method.

Key Concepts

Rationalizing the Denominator

Rationalizing the denominator is a process used to eliminate radicals (like square roots) from the denominator of a fraction. This makes the expression simpler and more standardized for further calculations. The key idea is to multiply the numerator and denominator by a radical that will clear the square root in the denominator.

In essence, for an expression like \(\frac{1}{\sqrt{a}}\), we multiply both parts of the fraction by \(\sqrt{a}\). This gives us:

• \(\frac{\sqrt{a}}{a} \).

Notice how the denominator becomes a non-radical number, making it easier to manage mathematically. For example, Justin's solution to \(\frac{7}{2\sqrt{7}}\) was correct because rationalizing effectively simplified the expression to \(\frac{\sqrt{7}}{2}\). Here, removing the radical from the denominator was the essential part of the process.

Overall, rationalizing helps to simplify complex expressions and is a foundational practice for working with radicals in mathematics.

Square Roots

Square roots, denoted with the radical symbol \(\sqrt{}\), are mathematical operations that find a number which, when multiplied by itself, yields the original number. For instance, \(\sqrt{49} = 7\) because \(7 \times 7 = 49\).

When working with square roots, it's important to understand and utilize perfect squares — numbers like 1, 4, 9, 16, 25, which have integer square roots. Recognizing these can simplify calculations dramatically.

In division involving square roots, you may often encounter expressions like:

• \(\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}\) when \(b\) is a perfect square.

This property can help simplify the component under the radical further, which is why it's valuable when simplifying expressions. For the expression \(\frac{7}{2\sqrt{7}}\), recognizing \(7\) as \(\sqrt{49}\) allowed the problem to become more manageable.

Ultimately, understanding and applying square roots correctly can significantly ease the simplification processes involving radicals.

Radicals in Denominators

Handling radicals in denominators is crucial because mathematical expressions are often easier to work with when denominators are rational (non-radical) numbers. This is why steps in simplifications, such as in Justin's method, focus on this transformation.

In fractions, like \(\frac{7}{2\sqrt{7}}\), having a radical in the denominator isn't ideal. Consequently, rationalizing it not only simplifies the appearance but also aids in performing subsequent operations like addition or subtraction.

When encountering fractions with denominators like \(2\sqrt{5}\), the challenge arises because \(\sqrt{5}\) is irrational and does not resolve into a neat fraction. In these cases, the approach might change slightly. Instead of simplifying further using square roots, showcasing its behavior in operations or estimates may suffice until a potentially rationalizing factor can be identified.

So, while attempting to simplify \(\frac{7}{2\sqrt{5}}\), the limitation lies in realizing \(\sqrt{5}\) doesn't lead to a simple rational expression. Nonetheless, maintaining the skills to handle radicals is critical.