Subtraction of Square Roots
2026-02-28 11:14 Diff
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Last updated on August 30, 2025

The mathematical operation of finding the difference between two square roots involves the subtraction of square roots. This operation helps simplify expressions and solve problems involving square roots.

What is Subtraction of Square Roots?

Subtracting square roots involves simplifying the difference between two or more square root terms.

It requires ensuring that the square roots have the same radicand (the number inside the square root).

If the radicands are identical, you can subtract their coefficients.

Components involved in the subtraction of square roots include:

Radicands: These are the numbers under the square root symbol.

Coefficients: These are the numbers in front of the square root symbol.

How to do Subtraction of Square Roots?

When subtracting square roots, students should follow these rules:

Check for like radicands: Only square roots with identical radicands can be subtracted from one another.

Combine coefficients: Subtract the coefficients of like square roots.

Simplify result: If possible, simplify the resulting square root expression further.

Methods to do Subtraction of Square Roots

The following are methods for subtracting square roots:

Method 1: Direct Subtraction

For direct subtraction of square roots, follow these steps:

Step 1: Ensure the radicands are the same.

Step 2: Subtract the coefficients of the square roots.

Step 3: Simplify the expression if possible.

Example: Subtract √18 from 3√18

Step 1: Check radicands: both are √18.

Step 2: Subtract coefficients: 3 - 1 = 2.

Answer: 2√18

Method 2: Simplifying First Sometimes, it's easier to simplify square roots before subtracting:

Example: Subtract √50 from 5√2

Solution: Simplify √50 to 5√2, then subtract. 5√2 - 5√2 = 0

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Properties of Subtraction of Square Roots

Subtraction of square roots has characteristic properties:

Subtraction is not commutative In subtraction, changing the order of terms changes the result, i.e., A - B ≠ B - A

Subtraction is not associative Regrouping changes the result: (A - B) - C ≠ A - (B - C)

Subtracting zero from an expression leaves the expression unchanged

Subtracting zero from any expression results in the same expression: A - 0 = A

Tips and Tricks for Subtraction of Square Roots

Tips and tricks are useful for efficiently dealing with the subtraction of square roots. Some helpful tips are:

Tip 1: Make sure radicands are the same before subtracting coefficients.

Tip 2: Simplify square roots whenever possible to make subtraction easier.

Tip 3: Use the properties of square roots to simplify expressions before subtraction.

Ignoring unlike radicands

Ensure that the radicands are identical before subtracting coefficients. Subtracting square roots with different radicands leads to errors.

Problem 1

Check radicands: both are √12. Subtract coefficients: 5 - 1 = 4 Result: 4√12

Okay, lets begin

Subtract √45 from 2√45

Well explained 👍

Problem 2

Check radicands: both are √45. Subtract coefficients: 2 - 1 = 1 Result: √45

Okay, lets begin

Subtract 3√7 from 7√7

Well explained 👍

Problem 3

Both terms have the radicand √7. Subtract coefficients: 7 - 3 = 4 Result: 4√7

Okay, lets begin

Subtract 4√3 from 9√3

Well explained 👍

Problem 4

Both terms have the radicand √3. Subtract coefficients: 9 - 4 = 5 Result: 5√3

Okay, lets begin

Subtract 2√11 from 6√11

Well explained 👍

No, only square roots with the same radicands can be subtracted.

1.Is subtraction of square roots commutative?

No, the order of terms matters in subtraction; changing them changes the outcome.

2.What are like radicands?

Like radicands refer to square roots that have the same number under the square root symbol.

3.What is the first step in subtracting square roots?

The first step is to ensure that the radicands are identical before performing subtraction.

4.What is the best method for subtracting square roots?

The best method is to ensure radicands are the same and then subtract coefficients directly.

Common Mistakes and How to Avoid Them in Subtraction of Square Roots

Subtraction of square roots can be challenging and often leads to common mistakes. Being aware of these errors can help students avoid them.

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Hiralee Lalitkumar Makwana

About the Author

Hiralee Lalitkumar Makwana has almost two years of teaching experience. She is a number ninja as she loves numbers. Her interest in numbers can be seen in the way she cracks math puzzles and hidden patterns.

Fun Fact

: She loves to read number jokes and games.