Problem 2 - Intermediate Algebra

Question

Evaluate each of the following. For example, \(\sqrt{25}=5\). \(\sqrt{49}\)

Step-by-Step Solution

Answer

The square root of 49 is 7.

Step 1: Understand the Square Root

The square root operation finds a number which, when multiplied by itself, results in the given number under the square root sign. This is expressed as \( \sqrt{x} \), where \( x \) is the number under the square root.

Step 2: Identify the Number Under the Square Root

In this problem, \( 49 \) is the number under the square root sign. Our task is to find a number which, when squared (multiplied by itself), equals 49.

Step 3: Find the Square Root

Identify a number \( n \) such that \( n \times n = 49 \). We know from basic arithmetic that \( 7 \times 7 = 49 \). Thus, \( n = 7 \).

Step 4: State the Solution

Thus, the square root of \( 49 \) is \( 7 \). We write this as \( \sqrt{49} = 7 \).

Key Concepts

Arithmetic

Arithmetic is the branch of mathematics dealing with numbers and the basic operations between them, such as addition, subtraction, multiplication, and division. For students beginning their journey into mathematics, understanding arithmetic is crucial as it forms the foundation for more advanced topics.

To grasp arithmetic fully, you should focus on the basic operations:

• Addition: Combining two numbers to get a total.
• Subtraction: Finding the difference between two numbers.
• Multiplication: A shortcut for repeated addition of the same number.
• Division: Splitting a number into equal parts.

Having a solid understanding of arithmetic operations helps in evaluating mathematical expressions, especially when dealing with square roots, as seen in the exercise of finding the square root of 49.

Understanding Square Roots

Square roots can be a tricky concept at first. It is essentially a number that, when multiplied by itself, gives the original number. In the context of the exercise, finding \( \sqrt{49} \) means identifying what number equals 49 when squared.

When tackling square roots, it's useful to remember some perfect squares, which are numbers like 1, 4, 9, 16, 25, and so on. Each of these numbers has an integer as its root, such as:

• \( \sqrt{4} = 2 \)
• \( \sqrt{9} = 3 \)
• \( \sqrt{16} = 4 \)

These examples show how square roots reverse the squaring process. It’s also good to visualize square roots by considering a square with an area corresponding to the number under the root; \( \sqrt{49} \) represents the length of a side of a square with an area of 49.

Mathematical Operations

Mathematical operations like addition, subtraction, multiplication, division, and finding square roots allow us to manipulate numbers in various ways to solve problems. Each of these operations serves unique purposes and helps us explore different aspects of mathematics more deeply.

When it comes to finding a square root, the operation isn't about performing a direct calculation like addition or multiplication, but rather about identifying a relationship or property of numbers. This requires understanding the concept of exponents because square root is the inverse of squaring.

For example, \( 7 \times 7 = 49 \) shows the squaring process, and \( \sqrt{49} = 7 \) demonstrates its inverse, the square root. A clear grasp of these operations helps in solving problems efficiently and is essential as you tackle more complex mathematical tasks.