Problem 79 - Intermediate Algebra

Question

Simplify each expression, if possible. All variables represent positive real numbers. $$ \sqrt{32 b} $$

Step-by-Step Solution

Answer

The expression simplifies to \( 4 \sqrt{2b} \).

Step 1: Identify Perfect Squares

Identify the perfect square factor of the number 32. We know that 32 can be broken down into 16 and 2, where 16 is a perfect square.

Step 2: Rewrite the Radicand

Rewrite the radicand (the expression inside the square root) as a product of a perfect square and another number: \[ 32 = 16 imes 2 \] So, we have: \[ \sqrt{32b} = \sqrt{16 imes 2 imes b} \]

Step 3: Simplify the Square Root

Since 16 is a perfect square, we can simplify the square root: \[ \sqrt{16 \times 2 \times b} = \sqrt{16} \times \sqrt{2} \times \sqrt{b} \] This results in: \[ 4 \times \sqrt{2b} \]

Step 4: Final Simplified Expression

The simplified expression is: \[ 4\sqrt{2b} \]

Key Concepts

Perfect Squares

Perfect squares are numbers that can be expressed as the product of an integer with itself. For example, the number 16 is a perfect square because it can be written as \( 4^2 \). Recognizing perfect squares is essential when simplifying radical expressions, as it allows us to break down more complex numbers into easier components.

Some common perfect squares include:

• 1 (since \( 1^2 \)= 1)
• 4 (since \( 2^2 \)= 4)
• 9 (since \( 3^2 \)= 9)
• 16 (since \( 4^2 \)= 16)
• 25 (since \( 5^2 \)= 25)

When simplifying a square root, identify factors of the radicand that are perfect squares. In the expression \( \sqrt{32b} \), notice that 16, a perfect square, can be factored out from 32. This allows us to simplify the square root effectively.

Radicands

The term radicand refers to the number or expression found underneath a radical or square root symbol. In the expression \( \sqrt{32b} \), 32b is the radicand. Understanding how to manipulate radicands is critical in simplifying square root expressions.

To simplify a radicand, you often break it down into its component factors. Here, we look for factors that can make the process easier by involving perfect squares. For example:

• The radicand 32 can be rewritten as \( 16 \times 2 \), since 16 is a perfect square.
• This enables you to rewrite \( \sqrt{32b} \) as \( \sqrt{16 \times 2 \times b} \).

By working with radicands in this way, you prepare the groundwork for further simplification by separating and simplifying the perfect square portions.

Square Root Simplification

Square root simplification is the process of making square root expressions as simple as possible. This often involves breaking down the radicand into products of perfect squares and non-perfect squares. By simplifying these components separately, the entire expression becomes simplified.

In our example, once 16 is identified as a perfect square, the simplification proceeds as follows:

• Write \( \sqrt{32b} = \sqrt{16 \times 2 \times b} \).
• Recognize \( \sqrt{16} = 4 \) since 16 is a perfect square.
• This simplifies the expression to \( 4 \times \sqrt{2 \times b} \).

The final expression \( 4\sqrt{2b} \) is a simplified form of the original expression. By extracting the perfect square and multiplying it separately, the square root becomes much easier to work with.