Square Roots & Decimals

Unit: Squares, Cubes & Roots

Chapter: Square Roots & Decimals

Reference: – quare Roots of Decimal Numbers, Perfect Square Decimals, Finding Square Root of a Decimal by Prime Factorization, Finding Square Root of a Decimal by Division Method, Square Roots of Non-Perfect Square Decimals, Estimating Decimal Square Roots, Number of Decimal Places in Square Root, Real-Life Applications, Solved Examples, Odd-One-Out Problems, Common Mistakes

After studying this chapter, you should be able to understand:

• How to Find Square Roots of Decimal Numbers
• How to Identify Perfect Square Decimals
• How to Use Division Method for Decimal Square Roots
• How to Estimate Square Roots of Non-Perfect Square Decimals

Introduction to Square Roots & Decimals

Definition

Just like whole numbers, decimal numbers can also have square roots. A decimal is a perfect square decimal if its square root is a terminating decimal. For example, 0.25 is a perfect square decimal because √0.25 = 0.5. Not all decimals have neat square roots; many are irrational (non-terminating, non-repeating).

When we find square roots of decimals, we essentially ask:

"What decimal number, when multiplied by itself, gives this decimal?"

Understanding square roots of decimals is essential for working with measurements, areas, and scientific calculations.

Importance of Square Roots of Decimals

• Used in geometry (areas of squares with decimal side lengths)
• Used in physics and engineering measurements
• Used in finance (interest calculations)
• Essential for solving quadratic equations with decimal coefficients
• Helps in estimating square roots without a calculator

Example

√0.49 = 0.7 because 0.7 × 0.7 = 0.49
√0.0121 = 0.11 because 0.11 × 0.11 = 0.0121
√2 ≈ 1.4142 (non-terminating, irrational)

Subtopics

1. Square Roots of Perfect Square Decimals

A decimal is a perfect square decimal if its square root is a terminating decimal (or integer).

Rule: To find √(decimal), convert the decimal to a fraction, then take the square root of numerator and denominator separately, then convert back.

Example 1: √0.09 = √(9/100) = √9 / √100 = 3/10 = 0.3

Example 2: √0.0036 = √(36/10000) = √36 / √10000 = 6/100 = 0.06

Example 3: √1.44 = √(144/100) = √144 / √100 = 12/10 = 1.2

Example 4: √0.0004 = √(4/10000) = 2/100 = 0.02

Pattern to Remember:

Quick Rule: √(0.0a) where a is a perfect square? Careful with decimal places.

2. Number of Decimal Places in Square Root

When a decimal has an even number of decimal places, it may be a perfect square decimal.

Rule: If a decimal has 2n decimal places and the number formed by the digits is a perfect square, then its square root will have n decimal places.

Examples:

Important: 0.4 has only 1 decimal place, so it cannot be a perfect square decimal (√0.4 is irrational).

3. Finding Square Root of a Decimal by Prime Factorization Method

Steps:

1. Write the decimal as a fraction in simplest form
2. Find the square root of numerator and denominator separately
3. Convert back to decimal

Example 1: Find √0.0225

0.0225 = 225/10000 = (15²)/(100²)
√0.0225 = 15/100 = 0.15

Example 2: Find √0.000625

0.000625 = 625/1000000 = (25²)/(1000²)
√0.000625 = 25/1000 = 0.025

Example 3: Find √2.25

2.25 = 225/100 = (15²)/(10²)
√2.25 = 15/10 = 1.5

4. Finding Square Root of a Decimal by Division Method

The long division method for square roots works for decimals as well. Group digits after the decimal in pairs (00, 00, 00…).

Steps:

1. Group digits before and after decimal in pairs (from decimal point outward)
2. Find the largest number whose square is less than or equal to the first group
3. Bring down pairs of zeros after decimal as needed
4. Continue the division process to get decimal places

Example: Find √0.64

Group: . 64
First group after decimal: 64
Largest square ≤ 64 is 8² = 64
√0.64 = 0.8

Example: Find √0.9 (approx.)

Group: . 90 00 00
8² = 64 ≤ 90, remainder 26
Bring down 00 → 2600
Double the quotient (8×2=16). Find digit d such that 16d × d ≤ 2600 → d=1 (161×1=161) → remainder 99
So √0.9 ≈ 0.948… (actually √0.9 = 0.94868…)

5. Square Roots of Non-Perfect Square Decimals

Most decimals are not perfect squares. Their square roots are irrational (non-terminating, non-repeating). We can estimate them.

Estimating Square Roots of Decimals:

Method 1 – Use perfect square decimals as benchmarks

Example: Estimate √0.3

Perfect squares near 0.3: 0.25 (√0.25=0.5) and 0.36 (√0.36=0.6)
0.3 is closer to 0.25? Actually 0.3 – 0.25 = 0.05, 0.36 – 0.3 = 0.06, so slightly closer to 0.25
√0.3 ≈ 0.55 (actual 0.5477…)

Method 2 – Convert to fraction and estimate

√0.3 = √(3/10) = √3/√10 ≈ 1.732/3.162 ≈ 0.548

Method 3 – Use calculator (or approximate decimal multiplication)

0.55² = 0.3025 (a bit high), so √0.3 ≈ 0.547

6. Square Roots of Decimals Between 0 and 1

For decimals between 0 and 1, the square root is larger than the original number.

Examples:

• √0.25 = 0.5 (0.5 > 0.25)
• √0.09 = 0.3 (0.3 > 0.09)
• √0.01 = 0.1 (0.1 > 0.01)

Reason: When you multiply a number less than 1 by itself, you get an even smaller number.

7. Square Roots of Decimals Greater than 1

For decimals greater than 1 (but not whole numbers), the square root is smaller than the original number.

Examples:

• √1.44 = 1.2 (1.2 < 1.44)
• √2.25 = 1.5 (1.5 < 2.25)
• √3.24 = 1.8 (1.8 < 3.24)

Reason: For numbers greater than 1, squaring makes them larger.

8. Real-Life Applications of Decimal Square Roots

Solved Examples

Example 1 – Perfect Square Decimal: Find √0.36

Solution: √0.36 = √(36/100) = 6/10 = 0.6

Answer: 0.6

Example 2 – Perfect Square Decimal: Find √0.0025

Solution: 0.0025 = 25/10000 = 5²/100² → √0.0025 = 5/100 = 0.05

Answer: 0.05

Example 3 – Factor Method: Find √0.0144

Solution: 0.0144 = 144/10000 = (12²)/(100²) = 12/100 = 0.12

Answer: 0.12

Example 4 – Greater than 1: Find √2.56

Solution: 2.56 = 256/100 = (16²)/(10²) → √2.56 = 16/10 = 1.6

Answer: 1.6

Example 5 – Estimation: Estimate √0.5

Solution: Between √0.49=0.7 and √0.64=0.8, closer to 0.7. √0.5 ≈ 0.707

Answer: About 0.707

Common Mistakes to Avoid

Mistake 1 – Mismatching decimal places
Thinking √0.4 = 0.2 (wrong: 0.2² = 0.04).
Correct understanding: 0.4 has only 1 decimal place, so its square root cannot be a terminating decimal.

Mistake 2 – Ignoring zeros in decimal
0.0004 = 4/10000, √0.0004 = 2/100 = 0.02, not 0.2.
Correct understanding: Count total decimal places carefully.

Mistake 3 – Forgetting to convert to fraction
Instead of √0.49 = √49/√100 = 7/10 = 0.7, some incorrectly try direct division.
Correct understanding: Convert to fraction when possible.

Mistake 4 – Placing decimal incorrectly in the root
√0.0169 = 0.13 (2 decimal places in original → 1 decimal place in root? Actually 0.13² = 0.0169, so 2 decimals in original, 2 in root? Wait: 0.13 has 2 decimals, 0.0169 has 4 decimals.
Correct understanding: Original has 2n decimal places → root has n decimal places.

Mistake 5 – Thinking all decimals have square roots that are decimals
√0.5 is irrational (≈0.707), not a terminating decimal.
Correct understanding: Only perfect square decimals have terminating square roots.

Mistake 6 – Confusing √0.1 with 0.1
√0.1 ≈ 0.316, not 0.1.
Correct understanding: Square root of a decimal between 0 and 1 is larger than the decimal itself.

Quick Reference Summary

Perfect Square Decimal: A decimal with an even number of decimal places whose square root is a terminating decimal

Square Root of Decimal (Fraction Method): Write decimal as fraction, take √ of numerator and denominator

Number of Decimal Places: If decimal has 2n decimal places and is a perfect square, its square root has n decimal places

√ of decimals between 0 and 1: Result is larger than the original number

√ of decimals greater than 1: Result is smaller than the original number

Common Perfect Square Decimals:

• 0.01 → 0.1
• 0.04 → 0.2
• 0.09 → 0.3
• 0.16 → 0.4
• 0.25 → 0.5
• 0.36 → 0.6
• 0.49 → 0.7
• 0.64 → 0.8
• 0.81 → 0.9
• 1.00 → 1.0
• 1.21 → 1.1
• 1.44 → 1.2