Problem 35 - Algebra for College Students

Question

Change each radical to simplest radical form. \(\frac{3}{2} \sqrt{24}\)

Step-by-Step Solution

Answer

The simplest form of \( \frac{3}{2} \sqrt{24} \) is \( 3 \sqrt{6} \).

Step 1: Identify the Radical Component

The expression is \( \frac{3}{2} \sqrt{24} \). We need to simplify \( \sqrt{24} \).

Step 2: Prime Factorization of the Radicand

Break down the number 24 into its prime factors: 24 = 2 \times 2 \times 2 \times 3 or \( 24 = 2^3 \times 3 \).

Step 3: Simplify the Square Root

The square root of a product that involves powers can be simplified. Rewrite \( \sqrt{24} \) as \( \sqrt{2^3 \times 3} \). Since \( 2^3 = 2^2 \times 2 \), we can simplify \( \sqrt{2^2 \times 2 \times 3} \) to \( 2 \sqrt{2 \times 3} \). Thus, \( \sqrt{24} = 2 \sqrt{6} \).

Step 4: Substitute and Simplify the Expression

Substitute the simplified radical back into the original expression: The expression becomes \( \frac{3}{2} \times 2 \sqrt{6} \).Since the coefficient 2 cancels with the denominator, the expression simplifies to: \( 3 \sqrt{6} \).

Key Concepts

Prime Factorization

Prime factorization is a technique used to break down a number into the prime numbers that multiply together to create the original number. It's like finding the building blocks of a number. For the radicand 24, we use this method to simplify the radical. To find the prime factors, start dividing by the smallest prime number 2 and continue until you can't divide evenly anymore. For 24:

• 24 divided by 2 is 12
• 12 divided by 2 is 6
• 6 divided by 2 is 3
• 3 is a prime number, so we stop here

Putting it all together: 24 = 2 × 2 × 2 × 3, or written with exponents, 24 = 2^3 × 3. Understanding the prime factorization helps in simplifying radicals through square root operations.

Square Root Simplification

Once we have the prime factorization, the next step is to simplify the square root. Simplifying means reducing the expression so it's written in the simplest radical form. For a square root like \( \sqrt{24} \), look at the prime factors: \( 24 = 2^3 \times 3 \).
Break down this expression based on the square root properties:

• Think of \( 2^3 \) as \( 2^2 \times 2 \)
• Take the square root of \( 2^2 \), which is 2
• So \( \sqrt{2^2 \times 2 \times 3} = 2 \sqrt{2 \times 3} \)

Hence, \( \sqrt{24} = 2 \sqrt{6} \). This method lets you take out perfect squares from under the radical sign.

Radical Expressions

A radical expression involves roots, such as square roots. Simplifying these expressions requires understanding the properties of radicals and coefficients. Starting with our example: \( \frac{3}{2} \sqrt{24} \), it can be broken into simpler parts:
Initially, simplify the radical \( \sqrt{24} \) to \( 2 \sqrt{6} \). Replace the simplified form back into our expression:

• Substitute as \( \frac{3}{2} \times 2 \sqrt{6} \)
• The 2 cancels out with the denominator 2
• Leaving us with \( 3 \sqrt{6} \)

This final expression, \( 3 \sqrt{6} \), is simpler and easier to interpret. In handling radical expressions, always check for factors that can be simplified by taking square roots, and remember to cancel out any common terms afterward.