How to Simplify Square Roots Without Calculator

What is a square root?

The square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square roots of 9 are 3 and -3 because 3 × 3 = 9 and (-3) × (-3) = 9.

Square roots are often represented with the radical symbol (√). For instance, √9 = 3. When a number has two square roots (one positive and one negative), we typically refer to the principal (positive) square root unless specified otherwise.

Why simplify square roots?

Simplifying square roots involves expressing them in their most reduced form, typically as a product of a perfect square and another square root. This process makes calculations easier and helps in solving equations and comparing quantities.

For example, √36 can be simplified to 6 because 6 × 6 = 36. Similarly, √72 can be simplified to 6√2 because 6 × 6 = 36 and 36 × 2 = 72.

Simplifying square roots is particularly useful in:

• Algebraic expressions
• Geometry problems involving areas and volumes
• Solving equations
• Comparing square roots

Methods to simplify square roots

There are several methods to simplify square roots without a calculator. The most common approach involves factoring the number under the radical into perfect squares and other factors.

Step 1: Factor the number

Break down the number under the radical into its prime factors. For example, to simplify √50:

• Factor 50: 50 = 25 × 2
• 25 is a perfect square (5 × 5)
• So, √50 = √(25 × 2) = √25 × √2 = 5√2

Step 2: Identify perfect squares

Common perfect squares to look for include:

• 1 (1 × 1)
• 4 (2 × 2)
• 9 (3 × 3)
• 16 (4 × 4)
• 25 (5 × 5)
• 36 (6 × 6)
• 49 (7 × 7)
• 64 (8 × 8)
• 81 (9 × 9)
• 100 (10 × 10)

Step 3: Apply the square root properties

Remember these properties when simplifying square roots:

For example, √(18/50) = √18 / √50 = 3√2 / 5√2 = 3/5

Step 4: Rationalize denominators

When simplifying expressions with square roots in the denominator, multiply the numerator and denominator by the square root in the denominator to eliminate it.

For example, to simplify √(2/3):

• Multiply numerator and denominator by √3
• √(2/3) × √(3/3) = √6 / 3

Worked examples

Let's look at several examples to see how to simplify square roots step by step.

Example 1: Simplifying √72

1. Factor 72: 72 = 36 × 2
2. 36 is a perfect square (6 × 6)
3. √72 = √(36 × 2) = √36 × √2 = 6√2

Example 2: Simplifying √128

1. Factor 128: 128 = 64 × 2
2. 64 is a perfect square (8 × 8)
3. √128 = √(64 × 2) = √64 × √2 = 8√2

Example 3: Simplifying √(18/50)

1. Factor numerator and denominator: 18 = 9 × 2, 50 = 25 × 2
2. √(18/50) = √(9×2 / 25×2) = √9 / √25 × √2 / √2 = 3/5 × √2/√2
3. Simplify √2/√2 to 1
4. Final simplified form: 3/5

Example 4: Rationalizing √(2/3)

1. Multiply numerator and denominator by √3
2. √(2/3) × √(3/3) = √6 / 3

Common mistakes

When simplifying square roots, it's easy to make mistakes. Here are some common errors to avoid:

1. Incorrect factoring

Make sure you've correctly factored the number under the radical. For example, 50 is not 25 × 3, but 25 × 2.

2. Missing perfect squares

Don't forget to check for larger perfect squares. For instance, 100 is a perfect square, not just 1, 4, 9, 16, 25, 36, 49, 64, or 81.

3. Improper simplification

Avoid simplifying √(a + b) or √(a - b) unless a and b are perfect squares. For example, √(10) cannot be simplified further.

4. Forgetting to rationalize

When dealing with square roots in denominators, remember to rationalize them by multiplying numerator and denominator by the square root in the denominator.