Square roots of perfect squares

Key Notes :

🟩 Square Roots of Perfect Squares

💡 What is a Square Root?

The square root of a number is the value that, when multiplied by itself, gives the original number.

Example:

• √25 = 5 because 5 × 5 = 25 ✅

🧮 Perfect Squares

A perfect square is a number that is the square of a whole number.

Examples:

• 1, 4, 9, 16, 25, 36, 49, 64, 81, 100 …

Number	Perfect Square of
1	1²
4	2²
9	3²
16	4²
25	5²
36	6²
49	7²
64	8²
81	9²
100	10²

🧩 Symbol of Square Root

The symbol √ is called the radical sign.

Example:

• √49 = 7 ✨

🧠 Finding Square Roots of Perfect Squares

There are three methods to find square roots:

🔹 (a) Prime Factorization Method

• Express the number as a product of prime factors.
• Make pairs of the same factors.
• Take one factor from each pair and multiply them.

👉 Example:

• √144
• = √(2 × 2 × 2 × 2 × 3 × 3)
• = 2 × 2 × 3 = 12

🔹 (b) Repeated Subtraction Method

• Subtract consecutive odd numbers starting from 1 until you reach 0.
• The number of subtractions gives the square root.

👉 Example:

• 49 − 1 = 48
• 48 − 3 = 45
• 45 − 5 = 40
• 40 − 7 = 33
• 33 − 9 = 24
• 24 − 11 = 13
• 13 − 13 = 0

✔ There are 7 subtractions, so √49 = 7

🔹 (c) Estimation Method

• Used when the number is not a perfect square.
• Find the nearest perfect squares and estimate the root.
• 👉 Example: √50 is between √49 (=7) and √64 (=8).
  So √50 ≈ 7.1

🟦 Properties of Square Roots

1️⃣ √(a × b) = √a × √b

2️⃣ √(a/b) = √a ÷ √b

3️⃣ (√a)² = a

4️⃣ √(a²) = |a| (always positive)

🎯 Square Root of Zero and One

• √0 = 0
• √1 = 1

⚡ Fun Fact

The square root of a perfect square is always a whole number, but the square root of a non-perfect square is irrational.

🌈 Examples

Expression	Solution	Answer
√16	4 × 4 = 16	4
√121	11 × 11 = 121	11
√225	15 × 15 = 225	15
√400	20 × 20 = 400	20

💬 Remember

👉 The square root “undoes” squaring.

👉 Perfect squares have exact square roots.

👉 Non-perfect squares give decimal or irrational results.

Learn with an example

Let’s try some problems!✍️