5.1 Introduction | Class 8 Mathematics Notes

5.5 Square Roots

This section introduces the concept of square roots as the inverse operation of squaring. It begins with real-life situations where finding the side of a square or the length of a triangle side requires determining a number whose square is known.

The square root of a number is defined as the number which, when multiplied by itself, gives the original number. For example, since 3² = 9, the square root of 9 is 3. The section notes that every perfect square has two integral square roots: one positive and one negative (e.g., 9 and -9 are square roots of 81). However, for the purpose of this chapter, only the positive square root is considered and denoted by the symbol √.

The section presents a table showing numbers from 1 to 10 and their squares, along with the corresponding square roots.

It then explains the method of finding square roots through repeated subtraction of successive odd numbers starting from 1. For example, subtracting 1, 3, 5, 7, 9, ... from 25 successively results in zero at the fifth step, indicating that 25 is a perfect square and its square root is 5.

Next, the section introduces finding square roots through prime factorisation. It explains that the prime factors of a square number occur in pairs. By pairing the prime factors of a number, the square root can be found as the product of one factor from each pair. Examples include finding the square root of 324 and 256.

The section also discusses how to determine if a number is a perfect square by checking if all prime factors occur in pairs. If not, multiplying by the missing factors can yield a perfect square.

Finally, the section introduces the long division method for finding square roots of large numbers. It explains how to estimate the number of digits in the square root based on the number of digits in the square and demonstrates the step-by-step procedure with examples.

Exercises encourage students to practice finding square roots using repeated subtraction, prime factorisation, and the division method.

📊 Diagram: Figures illustrating the concept of square roots in squares and right triangles (Fig 5.1 and Fig 5.2); Tables showing prime factorisation; Stepwise long division method for √529 and √4096; See figure_20, figure_21, figure_22, figure_23, figure_24, figure_25.

🧪 Activity: Students practice finding square roots by repeated subtraction, prime factorisation, and long division methods.

🔗 Connection: This section connects to the next, which covers square roots of decimal numbers and further applications.