Square and Square Roots Lesson - Definition, Tables, Examples
Lesson Overview
- Square and Square Roots Table
- What Are Perfect Squares?
- How to Find the Square
- Properties of Square Numbers
- Solved Examples on Squares and Square Roots
- Square and Square Roots Assessments
What Are Squares and Square Roots?
A square is the result of multiplying a number by itself.
For example, 4 × 4 = 16, so 16 is the square of 4.
A square root is the number you multiply by itself to get the square.
For example, the square root of 16 is 4 because 4 × 4 = 16.
Square and Square Roots Table
| Number (n) | Square (n²) | Square Root (√n) |
| 1 | 1 | 1 |
| 2 | 4 | 1.41 |
| 3 | 9 | 1.73 |
| 4 | 16 | 2 |
| 5 | 25 | 2.24 |
| 6 | 36 | 2.45 |
| 7 | 49 | 2.65 |
| 8 | 64 | 2.83 |
| 9 | 81 | 3 |
| 10 | 100 | 3.16 |
| 11 | 121 | 3.32 |
| 12 | 144 | 3.46 |
| 13 | 169 | 3.61 |
| 14 | 196 | 3.74 |
| 15 | 225 | 3.87 |
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What Are Perfect Squares?
Perfect squares are numbers obtained by squaring whole numbers. Examples are:
- 1 is a perfect square because 1 × 1 = 1.
- 25 is a perfect square because 5 × 5 = 25.
- 64 is a perfect square because 8 × 8 = 64.
How to Find the Square
Finding the square of a number is simple. Follow these steps:
- Understand what a square means
- The square of a number is the result of multiplying that number by itself.
- For example, if the number is 3, its square is 3 × 3 = 9.
- Write the number twice
- Take the number and write it down two times as a multiplication.
- Example: To find the square of 5, write 5 × 5.
- Multiply the number by itself
- Perform the multiplication.
- Example: 5 × 5 = 25. So, the square of 5 is 25.
- Use a calculator if needed
- If the number is large, use a calculator to make the multiplication faster and more accurate.
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Properties of Square Numbers
| Property | Description | Example |
| Non-Negativity | Square numbers are always non-negative. | (−3)2 = 9 32 = 9 |
| Odd or Even Nature | Squares of even numbers are even; squares of odd numbers are odd. | 42=16 (even) 52 = 25 (odd) |
| End Digits | A square number ends in 0,1,4,5,6, or 9; never 2,3,7, or 8. | 62 = 36 72 = 49 |
| Triangular Number Connection | The sum of two consecutive triangular numbers is a square number. | T3+T4 = 6 + 10 = 16 = 42 |
| Difference Between Squares | Consecutive square numbers differ by consecutive odd numbers. | 52 − 42 = 25 − 16 = 9 (an odd number) |
| Sum of First n Odd Numbers | The sum of the first n odd numbers equals n2. | 1 + 3 + 5 = 9 = 32 |
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Solved Examples on Squares and Square Roots
Example 1: Find the square of 7.
Solution:
- The square of a number is obtained by multiplying the number by itself.
- Square of 7 = 7 × 7 = 49.
- Answer: 49
Example 2: Find the square root of 81.
Solution:
- The square root is the number which, when multiplied by itself, gives the square.
- We know 9 × 9 = 81.
- Therefore, the square root of 81 is 9.
- Answer: 9
Example 3: Check if 36 is a perfect square.
Solution:
- A number is a perfect square if it can be expressed as the square of a whole number.
- Find the square root of 36.
- 6 × 6 = 36, so the square root of 36 is 6.
- Since the square root is a whole number, 36 is a perfect square.
- Answer: Yes, 36 is a perfect square.
Example 4: Calculate the square of 15 and explain if it is even or odd.
Solution:
- Square of 15 = 15 × 15 = 225.
- 15 is an odd number, and the square of an odd number is always odd.
- Answer: 225 (odd number).
Example 5: Find the square root of 144 and verify the result.
Solution:
- The square root of 144 is the number that, when multiplied by itself, equals 144.
- 12 × 12 = 144, so the square root of 144 is 12.
- Verification: 12 × 12 = 144 (Correct).
- Answer: 12
Square and Square Roots Assessments
- Write the squares of numbers 1 to 15.
- Find the square roots of 1, 16, 81, and 100.
- Check whether 49 and 72 are perfect squares by finding their square roots.
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