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Mom's Basement University

Operations with Radicals

What You’ll Learn

Section titled “What You’ll Learn”

In this lesson you’ll learn how to perform addition, subtraction, multiplication, and division with radical expressions and when simplification is required.

The Concept

Section titled “The Concept”

Adding and Subtracting Radicals - you can only combine radicals that have the same index and same radicand (like terms). Simplify each radical first, then combine.

Multiplying Radicals - for radicals with the same index, multiply the numbers inside:

a⋅b=a⋅b\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}

Dividing Radicals - combine under one radical, then simplify. Rationalize the denominator if needed:

ab=ab\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}

Key rule: always simplify radicals as much as possible before combining them.

Worked Example

Section titled “Worked Example”

1. Add: 5√12 + 3√3

Simplify √12 first:

12=4⋅3=23\sqrt{12} = \sqrt{4 \cdot 3} = 2\sqrt{3}

Now combine:

5(23)+33=103+33=1335(2\sqrt{3}) + 3\sqrt{3} = 10\sqrt{3} + 3\sqrt{3} = 13\sqrt{3}

2. Multiply: (3 + √5)(4 − √5)

Use FOIL:

=3(4)+3(−5)+5(4)+5(−5)=12−35+45−5=7+5\begin{aligned} &= 3(4) + 3(-\sqrt{5}) + \sqrt{5}(4) + \sqrt{5}(-\sqrt{5}) \\[1em] &= 12 - 3\sqrt{5} + 4\sqrt{5} - 5 \\[1em] &= 7 + \sqrt{5} \end{aligned}

Notice that √5 · √5 = 5 (the radical disappears when you multiply a radical by itself).

3. Divide and rationalize: 6 / √8

Simplify √8 first:

68=622=32\frac{6}{\sqrt{8}} = \frac{6}{2\sqrt{2}} = \frac{3}{\sqrt{2}}

Rationalize by multiplying top and bottom by √2:

32⋅22=322\frac{3}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} = \frac{3\sqrt{2}}{2}

Real-World Application

Section titled “Real-World Application”

Operations with radicals are used in:

  • Construction and carpentry (diagonal distances, roof pitches)
  • Physics (pendulum period, velocity formulas)
  • Engineering (stress calculations, electrical resistance)
  • Design (scaling shapes, calculating exact lengths)

Example: finding the diagonal of a rectangular room with sides 12 ft and 16 ft uses √(12² + 16²) = √(144 + 256) = √400 = 20 ft.

Quiz

Section titled “Quiz”
Simplify and add: $4\sqrt{18} + 3\sqrt{2}$
Multiply: $\sqrt{6} \cdot \sqrt{24}$
Rationalize the denominator: $\frac{5}{\sqrt{3}}$
You can add or subtract radicals only when they have:
Multiply and simplify: $(\sqrt{5} + 2)(\sqrt{5} - 2)$