Problem 63 - Algebra for College Students

Question

Express each of the following in simplest radical form. All variables represent positive real numbers. \(\sqrt{16 x+48 y}\)

Step-by-Step Solution

Answer

The simplest radical form is \(4 \sqrt{x + 3y}\).

Step 1: Factor the expression under the square root

First, identify any common factors in the expression under the square root. The expression is \(16x + 48y\). We can notice that both terms have a common factor. Let's factor it out.

Step 2: Find the greatest common factor

The greatest common factor of \(16x\) and \(48y\) is \(16\). Factor out \(16\) from the expression: \[ 16x + 48y = 16(x + 3y) \] Now, the expression under the square root is \(16(x + 3y)\).

Step 3: Simplify the square root

Use the property of square roots that states \(\sqrt{a \, b} = \sqrt{a} \times \sqrt{b}\). Therefore, \[ \sqrt{16(x + 3y)} = \sqrt{16} \times \sqrt{x + 3y} \] Since \(\sqrt{16} = 4\), the expression simplifies to \[ 4 \sqrt{x + 3y} \]

Key Concepts

Greatest Common Factor

When simplifying expressions, finding the greatest common factor (GCF) is essential. The GCF is the largest number that can divide each term of the expression evenly. In our problem, the expression under consideration is \(16x + 48y\). To determine the GCF:

• Identify the individual coefficients, which are 16 and 48.
• List the factors of each: 16 has factors \(1, 2, 4, 8, 16\) and 48 has factors \(1, 2, 3, 4, 6, 8, 12, 16, 24, 48\).
• The highest number common to both lists is 16.

Therefore, the GCF is 16. By factoring this out from the original expression, you simplify future steps and make computations easier.

Factoring Expressions

Factoring expressions involves rewriting an expression as a product of its factors. This practice is crucial in algebra for simplifying expressions and solving equations. In the original problem, after identifying the GCF as 16, we can factor the expression:

• Write the expression as \(16(x + 3y)\).
• Here, 16 is the factor outside, leaving \(x + 3y\) inside the parentheses.

This process breaks down complex expressions into simpler parts. Factoring expressions is a valuable skill for solving equations and simplifying results in many mathematical problems.

Properties of Square Roots

Understanding the properties of square roots is key when simplifying expressions like radicals. The specific property used here is \(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\). This means that the square root of a product can be expressed as the product of the square roots.
In our example, after factoring, the expression becomes \(\sqrt{16(x + 3y)}\). Using the property:

• First, split into \(\sqrt{16} \times \sqrt{x + 3y}\).
• Calculate \(\sqrt{16}\), resulting in 4 (since 4 times 4 equals 16).

Thus, the expression simplifies to \(4 \sqrt{x + 3y}\). Knowing these properties helps in handling and breaking down complex radicals into more manageable forms.