Problem 14 - Elementary and Intermediate Algebra

Question

Simplify each expression. $$ \sqrt{28} $$

Step-by-Step Solution

Answer

\( \sqrt{28} = 2 \sqrt{7} \)

Step 1: Identify Perfect Square Factors

To simplify \( \sqrt{28} \), start by identifying the largest perfect square factor of 28. The factors of 28 are 1, 2, 4, 7, 14, and 28. Among them, 4 is the largest perfect square.

Step 2: Rewrite the Radical Using Perfect Square Factor

Express 28 as a product of its factors, where one factor is the perfect square identified in Step 1. That is, \( 28 = 4 \times 7 \). Therefore, we can rewrite the radical as \( \sqrt{28} = \sqrt{4 \times 7} \).

Step 3: Simplify the Radical

Use the property of square roots that states \( \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \). Apply this to \( \sqrt{4 \times 7} = \sqrt{4} \times \sqrt{7} \). Since \( \sqrt{4} = 2 \), this simplifies to \( 2 \sqrt{7} \).

Step 4: Write the Simplified Expression

The final simplified form of \( \sqrt{28} \) is \( 2 \sqrt{7} \).

Key Concepts

Perfect Squares

A perfect square is an integer that is the square of another integer. Think of it as a number that can be neatly arranged into a square shape. For example:

• 1 is a perfect square because it is 1 squared.
• 4 is a perfect square as it is 2 squared.
• 9 is a perfect square, being 3 squared.

You can see the pattern here. Perfect squares are results of multiplying a whole number by itself. Recognizing these numbers is crucial when simplifying radicals like \(\sqrt{28}\). In the original problem, we identified 4 as a perfect square because 4 equals 2 squared. Spotting such factors helps break down radicals into simpler components, making calculations easier.

Square Roots

The square root of a number is a value that, when multiplied by itself, gives the original number. For instance, the square root of 16 is 4 since 4 times 4 equals 16. Square roots are represented by the radical sign \(\sqrt{}\). In any radical expression, it’s important to identify if the number has any perfect square factors, as these simplify the expression.

In the example \(\sqrt{28}\), we break it down using its perfect square factor 4. By expressing 28 as \(4 \times 7\), we use the property \(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\) to simplify. Thus, \(\sqrt{28} = \sqrt{4} \times \sqrt{7} = 2 \sqrt{7}\). This simplification process is essential in algebra, assisting in dealing with complex expressions more easily.

Algebraic Expressions

Algebraic expressions can include numbers, variables, and arithmetic operations. Simplifying these expressions often involves reducing radicals, which necessitates recognizing perfect squares and computing square roots. When dealing with expressions like \(\sqrt{28}\), it’s crucial to rewrite the expression in its simplest form.

Start by identifying perfect square factors, then decompose the expression. After simplifying, you're left with an expression that maintains its original value while being easier to understand and use in further calculations.

• Rearranging terms and breaking down the radicals is a common approach.
• This includes using properties such as \(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\).

Understanding these methods is key in handling algebra problems, ensuring that expressions are efficient to work with in broader mathematical contexts.