Problem 8 - Fundamentals of Mathematics

Question

Determine the value of each expression. \(\sqrt{49}\)

Step-by-Step Solution

Answer

The value of \(\sqrt{49}\) is 7.

Step 1: Understand the Square Root

The square root of a number is a value which, when multiplied by itself, gives the original number. For example, if \(a^2 = b\), then \(a\) is the square root of \(b\).

Step 2: Recognize Perfect Squares

Determine if the number inside the square root, 49, is a perfect square. A perfect square is a number that can be expressed as the product of an integer with itself. 49 is a perfect square because it is equal to \(7 \times 7\).

Step 3: Calculate the Square Root

Since 49 is a perfect square and equals \(7 \times 7\), the square root of 49 is 7. This means \(\sqrt{49} = 7\).

Key Concepts

Perfect Squares

A perfect square is a special type of number. It is the result of multiplying an integer by itself. For example, the numbers 1, 4, 9, 16, and 25 are all perfect squares.When you see a number like 49, you can check if it is a perfect square by finding an integer that multiplies by itself to make 49. Here, 7 is such a number because:

• 7 times 7 equals 49 (\(7 \times 7 = 49\))

Recognizing perfect squares is helpful when calculating square roots. This makes math problems easier to solve and understand.To summarize, think of perfect squares as building blocks. They show how numbers fit together neatly through multiplication.

Multiplication

Multiplication is a basic arithmetic operation. It combines groups of equal sizes to form a total. When dealing with square roots, multiplication helps us find perfect squares. For instance, when we say 49 is the product of 7 multiplied by 7, we are using multiplication to break down 49:

• 7 times 7 equals 49

This step shows how multiplication plays a vital role in understanding square roots. In general, multiplication is useful for:

• Determining products of numbers
• Understanding how numbers relate through multiplication
• Checking and verifying perfect squares and their square roots

It's a key tool for solving problems that involve square roots and perfect squares.

Integers

Integers are whole numbers that can be positive, negative, or zero. They are fundamental in mathematics, especially when you are working with square roots.In the case of perfect squares, integers are used because perfect squares are formed by multiplying integers.For example:

• The integer 7 multiplied by itself gives the perfect square 49

This means that when we calculate \(\sqrt{49}\), we are looking for an integer which, when multiplied by itself, results in 49.Understanding integers is vital as they form the basis of arithmetic operations and are key in finding roots of numbers. They make math intuitive and structured, aiding in visualizing and grasping the concepts of perfect squares and square roots.