Problem 32 - Algebra 1: Concepts and Skills

Question

Simplify the expression. $$ \sqrt{196} $$

Step-by-Step Solution

Answer

The square root of 196 is 14.

Step 1: Identify perfect square factor

The number inside the square root, known as the radicand, is 196. We are supposed to find a number that can be multiply by itself to get 196. From the various perfect square numbers that we know, we can recall that 14 multiplied by itself gives 196. So, we can say that 196 is a perfect square number and its square root is 14.

Step 2: Calculate square root of 196

After identifying that 196 is a perfect square number, we just have to express it as product of 14 and 14. The square root of 196 is, hence, equal to 14

Key Concepts

Perfect Square Numbers

To fully grasp the concept of simplifying square roots, it's essential to understand what perfect square numbers are. A perfect square number is the product of a number multiplied by itself. For instance, if you take the number 4 and multiply it by itself, you get 16, which makes 16 a perfect square number. Here's a breakdown of some common perfect square numbers:

• \textsf{1 is a perfect square because}
• \textsf{4 is a perfect square since it's} \(2 \times 2\).
• \textsf{9 is a perfect square because it's} \(3 \times 3\).
• \textsf{16 is a perfect square since it's} \(4 \times 4\).

By being familiar with perfect square numbers, you can rapidly identify and simplify square roots, thereby making it easier to simplify algebraic expressions that include them.

Radicand

A radicand is the number or expression inside a square root symbol. The process of simplifying a square root often involves your interaction with the radicand. For example, when you see \(\sqrt{196}\), the number 196 is the radicand. Before you can simplify the square root, you need to determine if the radicand is a perfect square number or if it can be broken down into factors that include a perfect square. Understanding the radicand is crucial as it directly influences how you approach the square root calculation.

Square Root Calculation

Calculating a square root can seem like a daunting task, but when you are dealing with a perfect square number, it's relatively straightforward. For instance, to calculate the square root of 196, you search for a number that when multiplied by itself gives you 196. In this case, the number is 14. Mathematically, this is represented as \(\sqrt{196} = 14\) because \(14 \times 14 = 196\). For square root calculations that don't result in whole numbers, you might need to use a calculator or learn more advanced methods such as prime factorization to simplify them.

Algebraic Expressions

Algebraic expressions are the cornerstone of algebra and include numbers, variables, and arithmetic operations. Simplifying square roots often involves working with these expressions. For example, an algebraic expression might have a term under a square root that needs to be simplified to solve an equation or to simplify the expression itself. Using the knowledge of perfect square numbers and understanding the radicand helps you to manipulate these algebraic expressions to reach a simpler form or to solve an equation.