Problem 24 - Elementary and 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 need to find the square root of 49. The square root operation asks which number, when multiplied by itself, gives the original number, which is 49 in this case.

Step 2: Recall Perfect Squares

Recall that a perfect square is a number that is the result of an integer multiplied by itself. List some perfect squares: 1, 4, 9, 16, 25, 36, 49, etc.

Step 3: Identify the Square Root

Find which integer squared gives 49. From the list of perfect squares, notice that 49 is a perfect square.

Step 4: Calculate the Square Root

Identify the integer corresponding to 49 from the perfect squares. Since \(7 \times 7 = 49\), the square root of 49 is 7.

Key Concepts

Understanding Perfect Squares

Perfect squares are numbers obtained by multiplying an integer by itself. They play a crucial role in understanding square roots. A perfect square is a result of such multiplication, resulting in numbers like 1, 4, 9, and so on. For example, if we multiply 3 by itself, we get 9, making it a perfect square. These numbers are called 'squares' because they can be arranged into perfect geometric squares visually.

• Perfect squares make calculating square roots more straightforward.
• Recognizing perfect squares helps identify numbers that have precise whole-number roots.
• When a number is a perfect square, its square root will be an integer.

Understanding perfect squares is foundational in mathematics because they simplify the process of finding square roots. Recognizing these special numbers can save time and ensure accuracy in mathematical operations involving square roots.

Importance of Integer Multiplication

Integer multiplication is a critical concept in mathematics. It is the process of multiplying two whole numbers, which always results in another integer. This principle is especially important when dealing with perfect squares and square roots.

• In the case of perfect squares, integer multiplication is used to create these special numbers. For instance, multiplying 6 by 6 gives 36, a perfect square.
• A solid understanding of integer multiplication aids in both verifying perfect squares and solving for square roots.
• As integers have no fractional or decimal component, their multiplication ensures the result will be a whole number, simplifying the identification of perfect squares.

Being comfortable with integer multiplication is essential for students working with square roots. It helps check solutions and understand the relationships between different mathematical operations.

Basic Mathematical Operations and Their Role

Mathematical operations such as addition, subtraction, multiplication, and division form the backbone of more complex concepts like evaluating square roots. Specifically, multiplication is crucial when dealing with square roots and perfect squares.

• Multiplication helps to identify perfect squares, which simplifies determining the square root of a number.
• Breaking arithmetic operations into fundamental steps can make complicated problems more manageable.
• Understanding these operations is necessary to grasp advanced mathematical concepts, as they often rely on these basics.

A strong grasp of basic mathematical operations enables students to tackle more challenging mathematical problems with confidence. They provide the toolkit needed to understand how numbers relate to each other, a skill that is highly beneficial in various areas of math.