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Chapter 8: Problem 19

Find the following roots. See Example 4. $$ \sqrt{49} $$

Short Answer

Expert verified
The square root of 49 is 7.

Step by step solution

01

Understanding the Problem

We need to find the square root of 49. This involves determining which number, when multiplied by itself, equals 49.
02

Identify Perfect Squares

Recall that a perfect square is a number that has an integer as its square root. 49 is a known perfect square.
03

Calculate the Square Root

To find the square root of 49, determine which integer multiplies by itself to produce 49. Since 7 脳 7 = 49, we conclude that the square root of 49 is 7.
04

Final Answer Verification

Multiply the result, 7, by itself to verify: 7 脳 7 = 49. This confirms our square root is correct.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Perfect Squares
A perfect square is a number that is the result of an integer multiplied by itself. These numbers are significant because their square roots are always whole numbers, or integers. Recognizing perfect squares can simplify many math problems. Some familiar perfect squares include:
  • 1 (since 1 x 1 = 1)
  • 4 (since 2 x 2 = 4)
  • 9 (since 3 x 3 = 9)
  • 16 (since 4 x 4 = 16)
  • 25 (since 5 x 5 = 25)
  • 36 (since 6 x 6 = 36)
  • 49 (since 7 x 7 = 49)

Grasping the concept of perfect squares can greatly assist in tasks such as simplifying square roots or solving quadratic equations. Keep in mind that as numbers get larger, it can become harder to quickly identify if they are perfect squares, but practice makes this easier.
Integer Multiplication
Integer multiplication is the process of finding the total when you have a certain number of equal groups of something. This is essential when dealing with perfect squares. To understand this, let's look at multiplying the integer 7 by itself, which is key to understanding why 49 is a perfect square. Imagine you have 7 rows of 7 apples each. The total number of apples will be:
  • 7 x 7 = 49

This result demonstrates how multiplication of a number by itself works. It solidifies understanding by confirming that multiplication of integers not only creates perfect squares but also helps verify if a number is indeed a perfect square.
Square Root Calculation
Calculating square roots involves determining what number times itself equals the given number. For perfect squares like 49, this process can be straightforward. The square root of a number is essentially the length of one side of a square with that number as its area. Let's explore:
  • The square root of 49 is found by identifying which integer times itself equals 49.
  • Here, 7 x 7 = 49, indicating that the square root is 7.

Verification is simple: when you multiply 7 by itself, the product is indeed 49, confirming the calculation is accurate. While perfect squares have whole numbers as roots, non-perfect squares will result in non-integer square roots. Thus, knowing how to calculate and verify square roots is crucial for mathematical accuracy.

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