Subtraction of Roots
2026-02-28 12:50 Diff
https://brightchamps.com/en-us/math/numbers/subtraction-of-roots

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Last updated on August 6, 2025

The mathematical operation of finding the difference between two square roots is known as the subtraction of roots. It helps simplify expressions and solve problems that involve square roots, constants, and arithmetic operations.

What is Subtraction of Roots?

Subtracting roots involves finding the difference between two square root expressions. It requires that the radicands (the numbers under the square root) are the same to directly subtract the roots. When the radicands are different, the expressions must be simplified or approximated using numerical values. There are three components to consider: Coefficients: These are constant values that multiply the roots, like 3√2 or -5√3. Radicands: These are the numbers inside the square root, such as 2 in √2 or 3 in √3. Operators: For subtraction, the operator is the minus (-) symbol.

How to do Subtraction of Roots?

When subtracting roots, students should follow these steps: Simplify: Simplify the square roots if possible by factoring out perfect squares. Combine like radicals: Only roots with the same radicand can be combined. Subtract their coefficients. Approximation: If the radicands are different and cannot be simplified, approximate their numerical values for subtraction.

Methods to do Subtraction of Roots

The following are the methods for subtraction of roots: Method 1: Simplification Method To apply this method, follow these steps: Step 1: Simplify each square root by factoring out perfect squares. Step 2: Combine like radicals by subtracting the coefficients. Example: Subtract √50 from 3√50. Step 1: Simplify: √50 = √(25×2) = 5√2 Step 2: Combine: 3(5√2) - 5√2 = 15√2 - 5√2 = 10√2 Method 2: Numerical Approximation When the roots cannot be simplified with common radicands, approximate their values. Example: Subtract √7 from 2√3. Solution: Approximate √7 ≈ 2.65 and √3 ≈ 1.73, then calculate 2(1.73) - 2.65 ≈ 3.46 - 2.65 ≈ 0.81

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Properties of Subtraction of Roots

The subtraction of roots has characteristic properties, including: Subtraction is not commutative: In roots subtraction, changing the order changes the result, i.e., √A - √B ≠ √B - √A. Subtraction is not associative: Changing the grouping of roots changes the result. (√A − √B) − √C ≠ √A − (√B − √C) Subtraction involves the addition of the opposite: Subtracting roots is like adding the negative, √A − √B = √A + (−√B). Subtracting zero from a root leaves it unchanged: √A - 0 = √A.

Tips and Tricks for Subtraction of Roots

Here are some useful tips for subtracting roots efficiently: Tip 1: Always simplify each root before attempting subtraction. Tip 2: Look for perfect square factors to simplify roots easily. Tip 3: If the radicands are different and cannot be simplified, consider numerical approximation for quick results.

Forgetting to simplify roots

Students often forget to simplify square roots before subtracting. Always simplify to find common radicands.

Problem 1

Use the simplification method, 4√18 − √18 = (4 - 1)√18 = 3√18

Okay, lets begin

Subtract 5√24 from 8√6

Explanation

Approximately 2.98

Well explained 👍

Problem 2

Use the numerical approximation method, Approximate √24 ≈ 4.9 and √6 ≈ 2.45, 5(4.9) - 8(2.45) ≈ 24.5 - 19.6 ≈ 4.9

Okay, lets begin

Subtract √32 from √50

Well explained 👍

Problem 3

Use the simplification method, √50 = √(25×2) = 5√2 √32 = √(16×2) = 4√2 5√2 - 4√2 = √2

Okay, lets begin

Subtract 2√45 from 9√20

Explanation

Approximately 8.8

Well explained 👍

Problem 4

Approximate √45 ≈ 6.7 and √20 ≈ 4.47, 2(6.7) - 9(4.47) ≈ 13.4 - 40.23 ≈ -26.83

Okay, lets begin

Subtract √72 from 3√18

Well explained 👍

No, roots with different radicands must be simplified or approximated before subtraction.

1.Is subtraction commutative for roots?

No, changing the order of roots changes the outcome.

2.What is a radicand?

A radicand is the number or expression inside the square root symbol.

3.What is the first step in subtracting roots?

The first step is to simplify each root by factoring out any perfect squares.

4.What method is used for subtracting roots?

The simplification method and numerical approximation are used for subtracting roots.

Common Mistakes and How to Avoid Them in Subtraction of Roots

Subtracting roots can be challenging, leading 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.