Square Roots - Seventh Grade

Perfect Squares & Estimation Methods

1. Understanding Square Roots

Definition

A square root of a number is a value that,

when multiplied by itself, gives the original number

√a

Read as "square root of a"

If √a = b, then b × b = a

or b² = a

Key Terms

Radical Symbol (√): The symbol used to denote square root

Radicand: The number under the radical symbol

Perfect Square: A number whose square root is a whole number

2. Perfect Squares

What is a Perfect Square?

A perfect square is a number that can be

expressed as the product of an integer by itself

• The square root of a perfect square is always a whole number

• Examples: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100...

Perfect Squares from 1 to 15

Number (n)	Square (n²)	Square Root (√n²)
1	1	√1 = 1
2	4	√4 = 2
3	9	√9 = 3
4	16	√16 = 4
5	25	√25 = 5
6	36	√36 = 6
7	49	√49 = 7
8	64	√64 = 8
9	81	√81 = 9
10	100	√100 = 10
11	121	√121 = 11
12	144	√144 = 12
13	169	√169 = 13
14	196	√196 = 14
15	225	√225 = 15

Examples

Example 1: Find √64

What number multiplied by itself equals 64?

8 × 8 = 64

Answer: √64 = 8

Example 2: Find √144

What number multiplied by itself equals 144?

12 × 12 = 144

Answer: √144 = 12

3. Estimating Square Roots

Why Estimate?

Not all numbers are perfect squares!

• √50 is NOT a whole number

• We can ESTIMATE its value

• Find which two perfect squares it falls between

Method: Between Two Perfect Squares

Step 1: Find the perfect squares BELOW and ABOVE the number

Step 2: Find the square roots of those perfect squares

Step 3: The square root lies BETWEEN those two values

Step 4: Estimate based on how close it is to each perfect square

Example 1: Estimate √50

Step 1: Find perfect squares around 50

49 < 50 < 64

Both 49 and 64 are perfect squares

Step 2: Find their square roots

√49 = 7 and √64 = 8

Step 3: √50 is between 7 and 8

7 < √50 < 8

Step 4: Refine estimate

50 is closer to 49 than to 64

So √50 is closer to 7

Estimate: √50 ≈ 7.1 (actual: 7.07)

Example 2: Estimate √80

Step 1: Find perfect squares around 80

64 < 80 < 81

Step 2: Find their square roots

√64 = 8 and √81 = 9

Step 3: √80 is between 8 and 9

8 < √80 < 9

Step 4: Refine estimate

80 is very close to 81

So √80 is closer to 9

Estimate: √80 ≈ 8.9 (actual: 8.94)

Example 3: Estimate √20

Perfect squares: 16 < 20 < 25

Square roots: √16 = 4 and √25 = 5

Therefore: 4 < √20 < 5

20 is closer to 16, so √20 is closer to 4

Estimate: √20 ≈ 4.5 (actual: 4.47)

4. Practice Problems

Finding Square Roots of Perfect Squares

Problem	Answer
√36	6
√100	10
√121	11
√169	13
√225	15

Estimating Square Roots

Problem	Between	Estimate
√30	5 and 6	≈ 5.5
√70	8 and 9	≈ 8.4
√90	9 and 10	≈ 9.5
√150	12 and 13	≈ 12.2

Quick Reference: Square Roots

Concept	Formula/Rule
Square Root Definition	If √a = b, then b² = a
Perfect Square	n² where n is a whole number
Estimating Method	Find between two perfect squares
Special Property	√(a²) = a (for positive a)