Problem 46 - A Graphical Approach to College Algebra

Question

If possible, simplify each radical expression. Assume that all variables represent positive real numbers. $$\sqrt{\frac{5}{3 p}}$$

Step-by-Step Solution

Answer

The simplified form is \( \frac{\sqrt{5}}{\sqrt{3p}} \).

Step 1: Identify the Expression

We start with the expression \( \sqrt{\frac{5}{3p}} \). Our goal is to simplify this radical expression as much as possible, given that all variables represent positive real numbers.

Step 2: Apply the Square Root to Numerator and Denominator

The properties of radicals allow us to separate the square root of a fraction into the root of the numerator over the root of the denominator. Thus, \( \sqrt{\frac{5}{3p}} = \frac{\sqrt{5}}{\sqrt{3p}} \).

Step 3: Simplify the Denominator

Next, simplify \( \sqrt{3p} \). Applying the property of square roots on the product, we have \( \sqrt{3p} = \sqrt{3} \cdot \sqrt{p} \). This gives us \( \frac{\sqrt{5}}{\sqrt{3} \cdot \sqrt{p}} \).

Step 4: Rationalize the Denominator (Optional)

If needed, multiply the numerator and denominator by \( \sqrt{3p} \) to rationalize the denominator: \( \frac{\sqrt{5}}{\sqrt{3} \cdot \sqrt{p}} \times \frac{\sqrt{3p}}{\sqrt{3p}} = \frac{\sqrt{5} \cdot \sqrt{3p}}{3p} \). However, since \( \sqrt{5} \cdot \sqrt{3p} = \sqrt{15p} \), we now have \( \frac{\sqrt{15p}}{3p} \).

Key Concepts

Square Root

Understanding the square root is fundamental to simplifying radical expressions. The square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 9 is 3 because 3 times 3 is 9.When dealing with fractions, the square root of a fraction can be broken down into the square root of the numerator divided by the square root of the denominator. This means if you have \( \sqrt{\frac{5}{3p}} \), you can express it as \( \frac{\sqrt{5}}{\sqrt{3p}} \).This separation makes it easier to address each part of the fraction individually and apply the rules of radicals more effectively, leading to simpler expressions overall.

Rationalizing the Denominator

Rationalizing the denominator is an essential process when simplifying radical expressions. It involves removing the radical from the denominator of a fraction. This is achieved by multiplying both the numerator and the denominator by a value that will leave the denominator as a whole number when multiplied.For instance, if you have \( \frac{\sqrt{5}}{\sqrt{3} \cdot \sqrt{p}} \), you can multiply both the top and bottom of the fraction by \( \sqrt{3p} \). This gives you:

• Numerator: \( \sqrt{5} \cdot \sqrt{3p} = \sqrt{15p} \)
• Denominator: \( \sqrt{3} \cdot \sqrt{p} \cdot \sqrt{3p} = 3p \)

This results in the simplified form \( \frac{\sqrt{15p}}{3p} \), where the denominator is now free of radicals.Rationalizing the denominator can make calculations simpler and results neater, particularly when dealing with more complex arithmetic.

Radical Expressions

Radical expressions involve roots, such as square roots or cube roots, and can sometimes appear complex. However, by understanding the properties of roots and how they interact, simplifying these expressions becomes manageable.The key to working with radical expressions, like \( \sqrt{5} \) or \( \sqrt{3p} \), is to identify opportunities to break them down into simpler parts. For products within the root, use the property \( \sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b} \). This allows for separation and easier management of each component.Simplification often involves reducing the expression to its most straightforward form, which might require actions like rationalizing the denominator or combining like terms. An expression such as \( \sqrt{\frac{5}{3p}} \) provides an opportunity to practice these strategies effectively.By isolating the root in both the numerator and the denominator, and potentially rationalizing the denominator, you achieve a format that's clearer and often easier to work with in further calculations or simplifications.