Problem 24 - Intermediate Algebra

Question

Evaluate each square root without using a calculator. See Objective 1 and Example 1. $$ \sqrt{49} $$

Step-by-Step Solution

Answer

The square root of 49 is 7.

Step 1: Understand the Problem

We are asked to find the square root of 49, which means we have to find a number that when multiplied by itself equals 49.

Step 2: Recall the Definition of a Square Root

The square root of a number is the value that, when squared, results in the original number. For instance, \( x^2 = 49 \) implies \( x = \sqrt{49} \).

Step 3: Identify Perfect Squares

Recall the list of perfect squares. 49 is a perfect square because it equals \( 7^2 \). So, \( \, 7 \, \times \, 7 = 49 \).

Step 4: Conclude the Calculation

Since 49 is a perfect square and can be represented as \( 7^2 \), then the square root of 49 is 7.

Key Concepts

Perfect Squares

Understanding perfect squares is crucial in solving square root problems easily. Perfect squares are numbers that are the result of an integer multiplied by itself. For example, 4 is a perfect square because it can be expressed as \( 2 \times 2 \), which is \( 2^2 \). Recognizing perfect squares helps speed up the process of finding square roots without a calculator.

• Examples of perfect squares: \( 1^2 = 1 \), \( 2^2 = 4 \), \( 3^2 = 9 \), \( 4^2 = 16 \), \( 5^2 = 25 \), and so on.
• Importance: Knowing these can assist in quickly identifying square roots. For instance, if you see 49, you should recognize it as a perfect square of 7 because \( 7 \times 7 = 49 \).

Listing and remembering these can reduce the need for guesswork or calculations with a calculator.

Understanding Objective 1

Objective 1 generally refers to the task's goal to evaluate square roots without a calculator through understanding basic mathematical principles. This objective is not just about computing but also about comprehending the underlying concepts.

• Focus: Developing mental math skills and ensuring students can work with numbers effortlessly.
• Application: Using perfect squares to determine square roots swiftly is part of this objective. For example, when confronted with \( \sqrt{49} \), rather than estimating, knowing \( 49 = 7^2 \) directly gives us 7.

This practice cultivates deeper numerical intuition, which is invaluable not only in academic settings but also in everyday life situations involving calculations.

Illustration: Example 1

Let's explore Example 1 to see square roots in action. Given the problem \( \sqrt{49} \), we’ll break it down:

• Step 1: Understand the problem. We need to find a number that, when multiplied by itself, gives 49.
• Step 2: Recall the concept of square roots. The square root of a number is the number that produces the original number when squared.
• Step 3: Identify if 49 is a perfect square. It is because \( 7^2 = 49 \).
• Conclusion: Hence, \( \sqrt{49} = 7 \).

This example exhibits the seamless application of recognizing perfect squares to quickly deduce the square root, eliminating the need for complex computations.