Adding and Subtracting Radicals

Learn how to combine like radicals using addition and subtraction, just like combining like terms.

Intermediate25 minLesson

Definition

To add or subtract radicals, they must be like radicals (same index and same radicand). Like radicals work just like like terms in algebra.

Like radicals have:

The same index (both square roots, both cube roots, etc.)
The same radicand (the number under the radical)

3\sqrt{5} + 7\sqrt{5} = 10\sqrt{5}

Think of it like combining apples: 3 apples + 7 apples = 10 apples.

Key Rule: You can only add or subtract the coefficients (numbers in front). The radical part stays the same.

a\sqrt{x} + b\sqrt{x} = (a + b)\sqrt{x}

a\sqrt{x} - b\sqrt{x} = (a - b)\sqrt{x}

Try it now

Which expression shows like radicals?

Worked Examples

Simplify: $4\sqrt{3} + 9\sqrt{3}$

Check if radicals are like

Both have $\sqrt{3}$ - same index (2) and same radicand (3) → Like radicals

Add the coefficients

$4 + 9 = 13$

Keep the radical part

$13 \cdot \sqrt{3}$ → $13\sqrt{3}$

Answer: $13\sqrt{3}$

Common Mistakes

Adding radicands: $\sqrt{3} + \sqrt{5} = \sqrt{8}$

Why it's wrong: You cannot add numbers under different radicals. This is like saying 1 apple + 1 orange = 2 apporanges!

Correct: $\sqrt{3} + \sqrt{5}$ cannot be simplified. They are unlike radicals.

Forgetting to simplify first: $\sqrt{8} + \sqrt{2} = \sqrt{8} + \sqrt{2}$

Why it's wrong: If you simplify $\sqrt{8} = 2\sqrt{2}$ first, you can combine: $2\sqrt{2} + \sqrt{2} = 3\sqrt{2}$

Correct: Always simplify radicals first, then check if they become like radicals.

Combining coefficients AND radicands: $3\sqrt{5} + 2\sqrt{5} = 5\sqrt{10}$

Why it's wrong: Only the coefficients change. The radicand stays the same.

Correct: $3\sqrt{5} + 2\sqrt{5} = 5\sqrt{5}$

Ignoring the coefficient of 1: $\sqrt{7} + 2\sqrt{7} = 2\sqrt{7}$

Why it's wrong: $\sqrt{7}$ means $1\sqrt{7}$ . So $1 + 2 = 3$ , not just $2$ .

Correct: $\sqrt{7} + 2\sqrt{7} = 1\sqrt{7} + 2\sqrt{7} = 3\sqrt{7}$

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Practice Problems

16 problems

Problem 1 of 16

Easy

Which expression shows like radicals?

Why It Matters

Adding and subtracting radicals appears throughout mathematics and science:

Geometry: Calculating perimeters when sides involve square roots, like $3\sqrt{2} + 5\sqrt{2}$ meters
Physics: Combining measurements involving roots in wave calculations
Engineering: Simplifying expressions in structural calculations
Finance: Some risk calculations involve combining radical expressions

Mastering this skill is essential for solving equations with radicals, working with the Pythagorean theorem, and advancing in algebra and calculus.

Real World Applications

Perimeter of a Triangle

When finding perimeters of shapes with sides expressed as radicals, you need to add them.

Example:

A triangle has sides $\sqrt{12}$ m, $\sqrt{27}$ m, and $\sqrt{48}$ m. The perimeter is $2\sqrt{3} + 3\sqrt{3} + 4\sqrt{3} = 9\sqrt{3}$ meters.

1Try It Yourself

A garden has a triangular shape with sides of $\sqrt{18}$ meters, $\sqrt{50}$ meters, and $\sqrt{8}$ meters.

What is the perimeter of the garden in simplest form?

Step 1: Write the mathematical expression

First simplify each side, then add:

Construction and Architecture

Architects often work with diagonal measurements that involve square roots.

Example:

Two roof sections have lengths $5\sqrt{3}$ meters and $2\sqrt{3}$ meters. Combined length: $7\sqrt{3}$ meters.

2Try It Yourself

A building has two diagonal supports: one is $\sqrt{75}$ meters and the other is $\sqrt{27}$ meters.

What is the total length of diagonal supports needed?

Step 1: Write the mathematical expression

Simplify each radical first:

Key Takeaways

1Like radicals have the same index and the same radicand (e.g., $3\sqrt{5}$ and $7\sqrt{5}$ )
2Add or subtract only the coefficients, keep the radical part unchanged
3Always simplify radicals first to find hidden like radicals (e.g., $\sqrt{8} = 2\sqrt{2}$ )
4Unlike radicals cannot be combined (e.g., $\sqrt{3} + \sqrt{5}$ stays as is)
5Think of radicals like variables: $3\sqrt{5} + 2\sqrt{5} = 5\sqrt{5}$ is like $3x + 2x = 5x$

Frequently Asked Questions

Yes, but simplify first!

\sqrt{4} = 2

and

\sqrt{9} = 3

, so

\sqrt{4} + \sqrt{9} = 2 + 3 = 5

. Note this is NOT

\sqrt{13}

Glossary

Radical: An expression containing a root symbol, such as $\sqrt{x}$ (square root) or $\sqrt[3]{x}$ (cube root)
Radicand: The number or expression under the radical sign. In $\sqrt{5}$ , the radicand is 5
Index: The small number indicating the type of root. Square roots have index 2 (often not written), cube roots have index 3
Like radicals: Radicals with the same index and the same radicand. $3\sqrt{7}$ and $5\sqrt{7}$ are like radicals
Coefficient: The number multiplied by the radical. In $4\sqrt{3}$ , the coefficient is 4
Simplify a radical: Rewrite with the smallest possible radicand, removing perfect square factors