Multiplying and Dividing Radical Expressions

Multiply and divide radical expressions and simplify. Math 20-1 Alberta Grade 11.

Lesson 5.3 of Radical Expressions and Equations in Math 20-1 — Alberta curriculum lessons.

What you'll learn in this lesson

• Multiplying Radicals (Single Term × Single Term). Learn the product rule for radicals and a clear step-by-step procedure: multiply coefficients, multiply radicands, then simplify the result by pulling out perfect-square factors.
• Dividing Radicals. Learn the quotient rule for radicals and practice dividing both the coefficients and the radicands, then simplifying the result — including cases where the denominator contains no radical.
• Multiplying with Distribution and FOIL. Apply the distributive property and FOIL to expand radical expressions involving brackets, recognizing when like radicals can be combined after expansion.
• Special Products and Conjugates. Master two powerful shortcuts: a square root multiplied by itself always gives the radicand, and multiplying conjugate pairs eliminates all radicals via the difference of squares pattern.

Inside the lesson — a free preview

The Basic Rule

Product Rule for Radicals. To multiply two radicals with the same index:

√(a) × √(b) = √(ab)

Multiply the coefficients together, multiply the radicands together, and put the result under one radical. Then simplify.

Important — Same Index Required. This rule only works when the two radicals have the same index. You can multiply √(3) × √(5) (both square roots), but you cannot directly combine √(3) × √(5) because they have different indices.

Step-by-Step Procedure

Procedure — Multiply r√(a) × s√(b). Step 1 — Signs: Determine the sign of the result (positive times positive stays positive; positive times negative becomes negative).

Step 2 — Coefficients: Multiply the coefficients (the numbers outside the radical). A radical with no written coefficient has an implied coefficient of 1.

Step 3 — Radicands: Multiply the radicands using √(a) · √(b) = √(ab).

Step 4 — Simplify: Simplify the resulting radical by pulling out any perfect-square factors.

Worked example: Worked Examples

Example 1 — Multiply √(3) × √(5). Step 1 — Coefficients: Both coefficients are 1.

Step 2 — Radicands: √(3) × √(5) = √(3 × 5) = √(15)

Step 3 — Simplify: Factors of 15 are 1, 3, 5, 15 — none (other than 1) are perfect squares. Already in simplest form.

Answer: √(15)

Example 2 — Multiply √(3) × √(8). Step 1 — Coefficients: Both coefficients are 1.

Step 2 — Radicands: √(3) × √(8) = √(3 × 8) = √(24)

Step 3 — Simplify √(24): Find the largest perfect-square factor of 24:

24 = 4 × 6

√(24) = √(4 × 6) = √(4) × √(6) = 2√(6)

Answer: 2√(6)

Example 3 — Multiply 5√(2) × 7√(3). Step 1 — Coefficients: 5 × 7 = 35

Step 2 — Radicands: √(2) × √(3) = √(6)

Step 3 — Combine: 5√(2) × 7√(3) = 35√(6)

Step 4 — Simplify: Factors of 6 are 1, 2, 3, 6 — no perfect-square factors. Already simplified.

Answer: 35√(6)

Example 4 — Multiply 4√(6) × 5√(2). Step 1 — Coefficients: 4 × 5 = 20

Step 2 — Radicands: √(6) × √(2) = √(12)

Step 3 — Combine: 4√(6) × 5√(2) = 20√(12)

Step 4 — Simplify √(12): Largest perfect-square factor of 12 is 4:

√(12) = √(4 × 3) = √(4) × √(3) = 2√(3)

So: 20√(12) = 20 × 2√(3) = 40√(3)

Answer: 40√(3)

Example 5 — Multiply 3√(6) × 2√(3). Step 1 — Coefficients: 3 × 2 = 6

Step 2 — Radicands: √(6) × √(3) = √(18)

Step 3 — Combine: 3√(6) × 2√(3) = 6√(18)

Step 4 — Simplify √(18): Largest perfect-square factor of 18 is 9:

√(18) = √(9 × 2) = √(9) × √(2) = 3√(2)

So: 6√(18) = 6 × 3√(2) = 18√(2)

Answer: 18√(2)

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