Problem 45 - Algebra A Combined Function

Question

Rationalize each numerator. Assume that all variables represent positive real numbers. \(\sqrt{\frac{5}{3}}\)

Step-by-Step Solution

Answer

The numerator is rationalized as \(\frac{5}{\sqrt{15}}\) or simplified to \(\frac{\sqrt{15}}{3}\).

Step 1: Identify the Numerator

In the expression \(\sqrt{\frac{5}{3}}\), the numerator is \(\sqrt{5}\) and the denominator is \(\sqrt{3}\). We need to rationalize the numerator \(\sqrt{5}\). This involves removing the square root from the numerator.

Step 2: Multiply by a Conjugate

To rationalize \(\sqrt{5}\), multiply the expression by \(\frac{\sqrt{5}}{\sqrt{5}}\): \[\sqrt{\frac{5}{3}} \times \frac{\sqrt{5}}{\sqrt{5}} = \frac{\sqrt{5} \times \sqrt{5}}{\sqrt{3} \times \sqrt{5}}\] This simplifies as it ensures the numerator becomes a rational number.

Step 3: Simplify the Numerator

Calculate \(\sqrt{5} \times \sqrt{5} = 5\). Thus, the rationalized expression becomes: \[\frac{5}{\sqrt{15}}\] Then, simplify further or rationalize as needed.

Step 4: Rationalize the Denominator (optional)

If further manipulation is needed, such as rationalizing the denominator \(\sqrt{15}\), multiply by a form of 1 using \(\sqrt{15}\): \[\frac{5 \times \sqrt{15}}{15} = \frac{5\sqrt{15}}{15\sqrt{15}} \]| Rationalizing \(\sqrt{15}\) yields: \[\frac{5\sqrt{15}}{15} = \frac{\sqrt{15}}{3}\] This presents the final simplified form if required.

Key Concepts

Simplifying Radicals

Simplifying radicals is a fundamental skill in algebra that involves removing any radicals (like square roots) to make an expression easier to understand. When you see an expression like \( \sqrt{\frac{5}{3}} \), notice that both the numerator and the denominator have square roots. However, the focus here is on simplifying the square root of a number, \( \sqrt{5} \), in the numerator.
To simplify a radical, you want to first check if the square root number can be broken down into smaller factors. For instance, if you have \( \sqrt{8} \), you can simplify it to \( \sqrt{4 \times 2} = 2\sqrt{2} \). But with \( \sqrt{5} \), because 5 is a prime number, there are no perfect squares in its factors, so it cannot be simplified further.
Often, simplifying also involves removing the radical completely from the numerator or denominator by multiplying or transforming the expression in certain ways, which we will explore further.

Multiplying by Conjugates

Multiplying by conjugates is a useful technique for rationalizing expressions. A conjugate involves taking a binomial expression \(a + b\) and creating another expression \(a - b\). When multiplied together, these expressions eliminate the radicals.
In the problem \( \sqrt{\frac{5}{3}} \), the step to rationalize the numerator \( \sqrt{5} \) involves multiplying the entire fraction by \( \frac{\sqrt{5}}{\sqrt{5}} \). Here, the conjugate trick isn't explicitly used in the exact form of a binomial but relies on the principle - primarily manipulating expressions to achieve rationality.
Multiplying the numerator by \( \sqrt{5} \):

• yields \( \sqrt{5} \times \sqrt{5} = 5 \)
• lets the denominator have the \( \sqrt{3} \times \sqrt{5} = \sqrt{15} \) as a result

This multiplication helps in simplifying the expression so the numerator is no longer in radical form, resulting in an expression that often is easier to work with.

Rationalizing Denominators

Rationalizing denominators is closely related to the technique of simplifying and serves a similar purpose but focuses on removing the radical from the denominator. In our example, the result of the initial rationalization led to \( \frac{5}{\sqrt{15}} \).
To ensure the expression is simplified completely, you may need to rationalize the denominator further. Multiply both the numerator and the denominator by \( \sqrt{15} \) to achieve this:

• The numerator \( 5 \times \sqrt{15} \) becomes \( 5\sqrt{15} \)
• The denominator \( \sqrt{15} \times \sqrt{15} \) changes to 15

The resulting expression is \( \frac{5\sqrt{15}}{15} \), which can still be simplified if possible. It's also critical in these steps to ensure that your work leads to the simplest equivalent form of the original expression. Rationalized expressions are often more useful and easier to understand or use in further calculations or applications.