Estimate positive and negative square roots
Key Notes :
🌟 Estimate Positive and Negative Square Roots
| 🔹 What is a Square Root? |
The square root of a number is a value that, when multiplied by itself, gives the original number.
- ✳️ Example: √25 = 5 because 5 × 5 = 25
| 🔹 Positive and Negative Square Roots |
Every positive number has two square roots:
- One positive ( + )
- One negative ( − )
✅ Example:
- √16 = ±4 → ( +4 and −4 ) because
- (4 × 4 = 16) and (−4 × −4 = 16)
| 🔹 Perfect Squares 🟩 |
Numbers like 1, 4, 9, 16, 25, 36, 49, 64, 81, 100 are perfect squares.
Their square roots are whole numbers.
- ✳️ Example: √49 = 7, √81 = 9
| 🔹 Non-perfect Squares 🟥 |
Some numbers (like 2, 3, 5, 6, 7, 8, 10, etc.) are not perfect squares.
Their square roots are irrational numbers (non-terminating decimals).
- ✳️ Example: √2 ≈ 1.41, √3 ≈ 1.73
| 🔹 Estimating Square Roots 🔢 |
If a number is not a perfect square, find two perfect squares between which the number lies.
📘 Steps:
- Find two nearest perfect squares.
- Find their square roots.
- The square root of your number will be between those two values.
✅ Example 1: Estimate √50
- 49 < 50 < 64
- √49 = 7 and √64 = 8
👉 So √50 ≈ 7.1
✅ Example 2: Estimate √8
- 4 < 8 < 9
- √4 = 2 and √9 = 3
👉 So √8 ≈ 2.8
| 🔹 Estimating Negative Square Roots 🚫 |
A negative square root just means the negative value of the positive root.
- ✳️ Example: √9 = ±3 → Positive root = 3, Negative root = −3
- ✳️ √50 ≈ ±7.1 → Positive ≈ +7.1, Negative ≈ −7.1
| 🔹 Square Roots of Negative Numbers ❌ |
You cannot find real square roots of negative numbers.
- ✳️ Example: √(−9) is not a real number.
| 📏 Helpful Tip: |
You can use a number line to visualize the position of square roots between two whole numbers!
| 🌈 Summary Table |
| Number | Perfect Squares Between | Estimated √ | Positive & Negative |
|---|---|---|---|
| 8 | 4 & 9 | ≈ 2.8 | ±2.8 |
| 20 | 16 & 25 | ≈ 4.5 | ±4.5 |
| 50 | 49 & 64 | ≈ 7.1 | ±7.1 |
| 90 | 81 & 100 | ≈ 9.5 | ±9.5 |
| 💡 Remember: |
Estimating square roots helps you understand the approximate value of irrational numbers — useful in geometry, algebra, and real-life measurements!
Learn with an example
Complete the following statement. Use the integers that are closest to the number in the middle.
( ) <-√15< ( )
For now, ignore the negative sign. Find the perfect squares that are just below and just above 15.
The perfect square just below 15 is 9:
√9 = 3
The perfect square just above 15 is 16:
√16 = 4
√15 is between √9 and √16 , so √15 is between 3 and 4 .
Finally, include the negative signs. Remember that with negative numbers, larger numbers like -√16 (if you ignore the minus sign) are less than smaller numbers.
-√16< – √15 <-√9 , so -4<-√15<-3
Complete the following statement. Use the integers that are closest to the number in the middle.
( ) <-√7< ( )
For now, ignore the negative sign. Find the perfect squares that are just below and just above 7.
The perfect square just below 7 is 4:
√4 = 2
The perfect square just above 7 is 9:
√9 = 3
√7 is between √4 and √9 , so √7 is between 2 and 3.
Finally, include the negative signs. Remember that with negative numbers, larger numbers like -√9 (if you ignore the minus sign) are less than smaller numbers.
-√9<-√7<-√4 , so -3<-√7<-2 .
Let’s try some examples! ✍️
