Problem 27 - Algebra 2 and Trigonometry

Question

In \(3-38\) , write each radical in simplest radical form. Variables in the radicand of an even index are non-negative. Variables occurring in the denominator of a fraction are non-zero. $$ \sqrt{\frac{15}{8 b^{3}}} $$

Step-by-Step Solution

Answer

\(\frac{\sqrt{15}}{2b\sqrt{b}}\) is the simplified form.

Step 1: Reduce the Fraction Under the Radical

Separate the fraction inside the radical into individual square roots for the numerator and the denominator: \[ \sqrt{\frac{15}{8b^3}} = \frac{\sqrt{15}}{\sqrt{8b^3}} \]

Step 2: Simplify the Denominator

Simplify the expression \(\sqrt{8b^3}\). First, note that \(8\) can be rewritten as \(2^3\). So, \[\sqrt{8b^3} = \sqrt{2^3b^3} = \sqrt{(2b)^3} = (2b)^{3/2} = 2b\sqrt{b} \]Thus, the complete expression is \[ \frac{\sqrt{15}}{2b\sqrt{b}} \]

Step 3: Express in Simplest Radical Form

Identify that the expression can be further simplified by using the properties of square roots. Combine the terms under the denominator's square root:\[ \frac{\sqrt{15}}{2b\sqrt{b}} = \frac{\sqrt{15}}{2b^{1.5}} = \frac{\sqrt{15}}{2b\sqrt{b}} \]The expression inside the denominator isn't in a simplified radical form than what is already obtained as \((2b\sqrt{b})\). Thus, the final simplest radical form remains as this.

Key Concepts

Fraction Under a Radical

Simplifying expressions that contain fractions under a radical involves separating the square roots, which results from the property: \( \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \). This technique allows us to handle the components in the numerator and the denominator individually, making it easier to simplify each part.
For example, consider the expression \( \sqrt{\frac{15}{8b^3}} \). By applying this property:

• Naturally, the expression becomes \( \frac{\sqrt{15}}{\sqrt{8b^3}} \).

This makes the next steps of simplifying the square root expressions simpler and clearer for each part of the fraction.
Breaking down the fraction in this way is a crucial step toward ensuring that the expression can be accurately and efficiently simplified.

Properties of Square Roots

Understanding the properties of square roots is vital in simplifying complex expressions. Particularly important is the property: \( \sqrt{x^2} = x \) and \( \sqrt{a^m} = a^{m/2} \), which are used to find simpler forms.
Applying these properties to our exercise, we address each part of \( \sqrt{8b^3} \). Here is how it works:

• Recognize that 8 can be expressed as \( 2^3 \). So when simplifying \( \sqrt{8} \), you get \( \sqrt{2^3} = 2^{3/2} = 2\sqrt{2} \).
• Similarly, simplify \( b^3 \). Using \( \sqrt{b^3} = b^{3/2} \), you convert this to \( b\sqrt{b} \).

This understanding of how to manage and simplify expressions containing square roots will be useful for identifying patterns and applying strategies to simplify more complex expressions.

Simplest Radical Form

The goal of simplifying a radical expression is to achieve its simplest radical form. This involves eliminating any unnecessary radicals and expressing the terms as simply as possible.
In our example, after splitting \( \sqrt{\frac{15}{8b^3}} \) into \( \frac{\sqrt{15}}{\sqrt{8b^3}} \), and further simplifying \( \sqrt{8b^3} \) into \( 2b \sqrt{b} \), we've reached the expression:

• \( \frac{\sqrt{15}}{2b\sqrt{b}} \)

This expression is already highly simplified. To check if it’s in its simplest radical form, verify:

• No perfect squares remain under any radicals.
• The radicals cannot be further simplified using basic properties.

Therefore, we conclude that \( \frac{\sqrt{15}}{2b\sqrt{b}} \) is not just simplified, but its simplest radical form. Understanding this ensures correct answers for similar problems in simplifying radical expressions.