Problem 9 - College Algebra

Question

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

Step-by-Step Solution

Answer

14

Step 1: Identify Perfect Square

The first step to simplifying \( \sqrt{196} \) is to determine if the number 196 is a perfect square. A perfect square is an integer that is the square of an integer. Check if there exists a whole number which, when multiplied by itself, equals 196.

Step 2: Find Square Root

Since 196 is indeed a perfect square, find the integer whose square is 196. We can see that \( 14 \times 14 = 196 \). Therefore, \( \sqrt{196} = 14 \).

Key Concepts

Perfect Squares

Understanding perfect squares is an essential part of simplifying square roots. A perfect square is a number that can be expressed as the product of an integer with itself. For example, numbers like 4, 9, 16, and 25 are perfect squares because:

• 4 = 2 × 2
• 9 = 3 × 3
• 16 = 4 × 4
• 25 = 5 × 5

Identifying perfect squares is useful because it allows us to determine if the square root of a number can be expressed as a whole number. When simplifying square roots, recognizing perfect squares helps to quickly identify numbers that can be simplified without using a calculator. Remember, if a number is a perfect square, its square root will be an integer.

Square Root Calculation

Square root calculation may seem intimidating at first, but it becomes straightforward once you grasp the basic concept. The square root of a number is essentially a value that, when multiplied by itself, gives the original number. In mathematical terms, the square root of a number \( x \) is represented as \( \sqrt{x} \).

For example, to calculate \( \sqrt{196} \), we need to determine if 196 is a perfect square (which we established it is). Once confirming that, we can find the integer that, when squared, equals 196. Since 14 × 14 = 196, it follows that \( \sqrt{196} = 14 \).
The process involves:

• Recognizing the number as a perfect square.
• Finding an integer that squares to give the original number.

For numbers that are not perfect squares, the square root can still be calculated, often resulting in a non-integer. In such cases, the number is often left in its root form or approximated to a decimal.

Integer Multiplication

Integer multiplication is a fundamental mathematics operation frequently used in the process of simplifying square roots.
An integer is a whole number, and multiplication simply involves combining these values through repeated addition. For example, when we multiply integers 14 × 14, we add 14 to itself 14 times, arriving at 196. This operation is crucial when checking if a number like 196 is a perfect square because it allows us to verify the original number can be produced by multiplying the integer by itself. Thus, consistent practice with integer multiplication helps reinforce mental math skills, making recognition of perfect squares more intuitive.