Class 8 CBSE – Mathematics
Squares & Square Roots
Topic Overview – In this topic you will learn how to square a number (multiply a number by itself) and how to extract the square root of a number (find a number that, when squared, gives the original number). These ideas are used everywhere in algebra, geometry, and applied maths.
1. Simple Explanation
| Concept | What it means | Quick Example |
|---|---|---|
| Square | Multiplying a number by itself. Denoted as (n^2). | (5^2 = 5 \times 5 = 25) |
| Square root | For a given number (x), a number (y) such that (y^2 = x). Denoted as (\sqrt{x}). | (\sqrt{25} = 5) (because (5^2 = 25)) |
Key Idea – Squaring and square‑rooting are inverse operations.
If you square a number and then take its square root, you get back the original number (for non‑negative numbers).
2. Key Points
| # | Concept | Important Notes |
|---|---|---|
| 1 | Perfect Squares | Integers whose square root is an integer. Examples: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, … |
| 2 | Square of a Binomial | ((a+b)^2 = a^2 + 2ab + b^2) ((a-b)^2 = a^2 - 2ab + b^2) |
| 3 | Difference of Squares | (a^2 - b^2 = (a+b)(a-b)). Useful for factoring. |
| 4 | Multiplication Property | (\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}) (only for (a,b ≥ 0)). |
| 5 | Dividing by a Square | (\frac{a^2}{b^2} = \left(\frac{a}{b}\right)^2). |
| 6 | Decimal Squares | Use the same rules; be careful with decimal places. |
| 7 | Finding Square Roots (without a calculator) | 1. Find the nearest perfect square 2. Estimate by linear interpolation or use the “average” method. |
| 8 | Using a Calculator | Enter the number and press the √ button. Remember to read the result correctly (e.g., √2 ≈ 1.4142). |
| 9 | Applications | • Area of a square (side × side) • Diagonal of a square (side × √2) • Pythagoras’ theorem in right‑angled triangles |
3. Worked Example
Problem
Find the square of 123 and then the square root of the result.
Step‑by‑Step
| Step | Operation | Calculation | Result |
|---|---|---|---|
| 1 | Square of 123 | (123^2) | 15 129 |
| 2 | Square root of 15 129 | (\sqrt{15,129}) | 123 |
Why It Works
- 123 × 123 = 15 129 (by the standard multiplication method or using the identity ((a+b)^2 = a^2+2ab+b^2) with a=120, b=3).
- Since 15 129 is a perfect square (it equals 123²), its square root is simply 123.
Quick Check
If you’re unsure, verify: (123 × 123 = 15 129).
Also, if you take the square root of 15 129 using a calculator, you’ll get 123.000… (rounded).
4. Quick Revision Summary (Bullet‑Points)
- Square: (n^2 = n × n).
- Square root: (\sqrt{x}) is a number whose square is (x).
- Perfect squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, …
- Binomial squares: ((a+b)^2 = a^2 + 2ab + b^2).
- Difference of squares: (a^2 - b^2 = (a+b)(a-b)).
- Multiplication rule: (\sqrt{ab} = \sqrt{a}\sqrt{b}).
- Division rule: (\dfrac{a^2}{b^2} = \Big(\dfrac{a}{b}\Big)^2).
- Estimating square roots: Find nearest perfect squares, then interpolate.
- Calculator tip: Use the √ button; press “C” or “AC” to clear.
- Applications: area of squares, Pythagoras, geometry of rectangles, etc.
5. Quick Practice Problems
- Calculate (37^2).
- Find (\sqrt{289}).
- Expand ((7-4)^2).
- Factorise (225 - 49).
- Estimate (\sqrt{50}) (to one decimal place).
*(Answers: 1. 1 369, 2. 17, 3. 9,