Problem 79 - Elementary and Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

\(\sqrt{32b} = 4\sqrt{2b}\).

Step 1: Break Down the Square Root

The expression under the square root is \(32b\). We can break it down as \(\sqrt{32b} = \sqrt{32} \times \sqrt{b}\). Since the variables represent positive real numbers, this breakdown is valid.

Step 2: Write 32 as a Product of Perfect Squares

The number 32 can be rewritten as a product of a perfect square, specifically, \(32 = 16 \times 2\). This allows us to use the property \(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\) for our next step.

Step 3: Simplify the Perfect Square

Substitute \(16 \times 2\) for \(32\) in the square root: \(\sqrt{32} = \sqrt{16 \times 2}\). Since 16 is a perfect square, we have \(\sqrt{16 \times 2} = \sqrt{16} \times \sqrt{2}\). Compute \(\sqrt{16} = 4\), so \(\sqrt{32} = 4\sqrt{2}\).

Step 4: Combine the Results

Now substitute back into the expression: \(\sqrt{32b} = \sqrt{32} \times \sqrt{b} = 4\sqrt{2} \times \sqrt{b}\). This simplifies to \(4\sqrt{2b}\). Therefore, \(\sqrt{32b} = 4\sqrt{2b}\).

Key Concepts

Perfect Squares

Understanding perfect squares is crucial when simplifying square roots. A perfect square is simply a number that can be expressed as the square of an integer. For example:

• The number 16 is a perfect square because it can be written as \(4^2\).
• Similarly, 9 is a perfect square since it equals \(3^2\).
• Numbers like 25 are also perfect squares, as they can be expressed as \(5^2\).

To simplify square roots, look for numbers under the square root sign that can be broken down into products of perfect squares. For instance, in the expression \(\sqrt{32}\), 32 can be rewritten as \(16 \times 2\). Knowing that 16 is a perfect square, we extract it: \(\sqrt{16} = 4\), thus simplifying our task.

Properties of Square Roots

Square roots possess several properties that make calculations easier. Recognizing these properties can greatly assist in simplification processes. Here are key properties:

• The square root of a product: \(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\). This allows you to split square roots into more manageable parts.
• The square root of a quotient: \(\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}\), useful when handling fractions.

Using these properties, you can simplify square roots by breaking down the expressions into smaller parts. For example, \(\sqrt{32b}\) is expressed as \(\sqrt{32} \times \sqrt{b}\). Applying these properties, specifically focusing on \(\sqrt{32}\), and identifying the perfect square factor \(16\) within it can make your simplification straightforward.

Positive Real Numbers

When working with square roots and variables, it's often mentioned that these variables are positive real numbers. But why is this important?Firstly, a real number is any value along the continuous number line, covering every possible number you might encounter in practical situations.

• Positive real numbers are those greater than zero.
• This assumption ensures that no negative numbers appear under the square root, as taking the square root of a negative number results in an imaginary number.

Thus, when you see an exercise specifying that variables are positive real numbers, it reassures us that the operations will not require dealing with complex numbers. This allows for straightforward simplification, like in our expression \(\sqrt{32b}\), where \(b\) is positive, leading smoothly to a simplified result: \(4\sqrt{2b}\).