Squares & Square Roots — Class 8

1. Definitions

Square: The square of a number a is a² = a × a. Example: 6² = 36.

Square root: The square root of a number n is a number r such that r² = n. We write r = √n. Example: √36 = 6. Every positive number has two square roots (positive & negative), but principal square root is positive.

2. Perfect Squares (first 30)

3. Methods to Find Square Roots

Prime Factorization Method

Factor the number into primes. If all prime powers are even, the number is a perfect square. Example: 144 = 2² × 3² → √144 = 2 × 3 = 6.

Long Division Method (for non-perfect squares)

Grouping digits in pairs from the decimal point and using the long division-like algorithm finds the square root digit by digit. Useful for larger numbers and to get decimal square roots.

Using Perfect Squares Table

Locate nearest perfect squares to approximate root. Example: √50 is between √49(=7) and √64(=8) → approx 7.07.

4. Shortcuts & Tricks

• If a number ends with 5, its square ends with 25. Shortcut: To square 25 → 2×3 = 6, append 25 → 625. (Rule: n5² = n(n+1)25)
• For quick estimation use nearest perfect squares.
• For checking if a large number is a perfect square: prime factorize or check last digit (perfect squares end with 0,1,4,5,6,9).
• To find √(a×b) where a and b are perfect squares: √a × √b.

5. Solved Examples

6. Practice Questions (with answers)

1. Find the square root of 784.
   Answer

   √784 = 28
2. Is 3600 a perfect square?
   Answer

   3600 = 60² → Yes, √3600 = 60
3. Find √(81 × 49).
   Answer

   √(81×49) = √81 × √49 = 9 × 7 = 63
4. Square 45 using shortcut (ends with 5).
   Answer

   45² → 4×5 = 20, append 25 → 2025

Practice the long division square root method on non-perfect squares to gain speed — it's often asked in competitive papers.

7. References

• NCERT Mathematics Class 8 — Chapter: Squares & Square Roots
• Practice worksheets and previous year sample papers