Relationship between squares and square roots

Key Notes:

🧮 Relationship Between Squares and Square Roots

🔹 What is a Square?

The square of a number is the product of the number multiplied by itself.

👉 Example:

• 5²=5×5=25

Symbol: n² (read as “n squared”)

✨ Examples:

Number	Square
2	4
3	9
4	16
10	100

🔹 What is a Square Root?

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

👉 Example:

• √25=5 because 5×5=25

Symbol:√​ (called the radical sign)

✨ Examples:

Number	Square Root
1	1
4	2
9	3
16	4
25	5

🔹 Relationship Between Squares and Square Roots

💡 Squares and square roots are inverse (opposite) operations.

Operation	Example
If you square a number and then take the square root, you get the original number.	√(52)=√25=5
If you take the square root first and then square, you also get the original number.	(√9)2=32=9

✅ In short: √a2=a and(√a)2=a

🔹 Perfect Squares

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

• 👉 Example: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100

🟢 Examples:

• 7²=49 → 49 is a perfect square
• 8²=64 → 64 is a perfect square

🔹 Important Facts to Remember

✨ The square of:

• A positive number ➜ is positive
• A negative number ➜ is also positive
  👉 Example: (−5)²=25
• The square root of a positive number ➜ has two values, one positive and one negative.
  👉 Example: √25=±5

🔹 Geometric Connection 📏

• The area of a square = side × side = side²
• The side length = √(area)

👉 If the area of a square = 49 cm²
Then, side = √49 = 7 cm

🔹 Quick Tricks 🪄

✅ Last digits of perfect squares: 0, 1, 4, 5, 6, 9

❌ A number ending in 2, 3, 7, or 8 cannot be a perfect square.

🧡 Summary Table

Concept	Example	Result
Square of 6	62	36
Square Root of 36	√36	6
Relationship	√62=6	Inverse operations

🧠 Remember:

“Squaring a number makes it bigger (except 0 and 1),
Taking the square root brings it back to the original!” 🌈

Learn with an example

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