Simplify Square Root on Calculator

Simplifying square roots is a fundamental math skill that helps you express square roots in their most reduced form. This guide explains how to simplify square roots using a calculator, including step-by-step instructions, examples, and a built-in calculator tool.

How to Simplify Square Roots

Simplifying a square root involves expressing it as a product of a perfect square and another square root. The general steps are:

Factor the number under the square root into its prime factors.
Identify pairs of identical factors (perfect squares).
Take one factor from each pair out of the square root.
Multiply the remaining factors inside the square root.

For example, √36 simplifies to 6 because 36 is a perfect square (6 × 6). For non-perfect squares, you can simplify by factoring out perfect squares.

Formula

√(a × b) = √a × √b

If a is a perfect square, √a = √(n²) = n

Step-by-Step Guide

Step 1: Factor the Number

Break down the number under the square root into its prime factors. For example, to simplify √72:

72 ÷ 2 = 36
36 ÷ 2 = 18
18 ÷ 2 = 9
9 ÷ 3 = 3
3 ÷ 3 = 1

So, 72 = 2 × 2 × 2 × 3 × 3

Step 2: Identify Perfect Squares

Look for pairs of identical factors. In the example above, there are three pairs of 2s and one pair of 3s.

Step 3: Take One from Each Pair

Take one factor from each pair out of the square root. For 72, this means taking two 2s and one 3.

Step 4: Multiply the Remaining Factors

Multiply the remaining factors inside the square root. For 72, this leaves one 2 inside the square root.

So, √72 = 2 × 3 × √(2) = 6√2

Tip

Always check if the number under the square root is a perfect square first. If it is, the square root is simply the square root of that perfect square.

Common Mistakes

When simplifying square roots, common errors include:

Forgetting to factor the number completely.
Not identifying all perfect square pairs.
Taking more factors out than are available.
Incorrectly multiplying the remaining factors.

Double-check your work by squaring the simplified form to ensure it equals the original number under the square root.

Examples

Example 1: Simplifying √48

Factor 48: 48 = 16 × 3 = 4 × 4 × 3

√48 = √(16 × 3) = √16 × √3 = 4√3

Example 2: Simplifying √108

Factor 108: 108 = 36 × 3 = 6 × 6 × 3

√108 = √(36 × 3) = √36 × √3 = 6√3

Example 3: Simplifying √200

Factor 200: 200 = 100 × 2 = 10 × 10 × 2

√200 = √(100 × 2) = √100 × √2 = 10√2

FAQ

Can I simplify square roots with variables?: Yes, the same principles apply. Factor the variable expression and look for perfect squares.
What if the number under the square root isn't a perfect square?: You can still simplify by factoring out the largest perfect square possible.
How do I know if I've simplified a square root correctly?: Square the simplified form and check if it equals the original number under the square root.
Can I use a calculator to simplify square roots?: Yes, calculators can help verify your simplification by computing the square root of the original and simplified forms.
What if the number under the square root is negative?: Square roots of negative numbers are not real numbers. They are complex numbers involving the imaginary unit i.