Simplify radical expressions using product relationships

Ontario Grade 11 Mathematics

Characteristics of Functions

MCR3U – A3.2 | Breaking Radicals Apart and Pulling Out Perfect Squares

Radicals appear throughout the MCR3U course — in quadratic models, trigonometric exact values, and sequences. Before you can work with them confidently, you need one key skill: simplifying them into their neatest possible form. This lesson focuses entirely on the product rule for radicals, the single most useful tool for simplification. You will start with a concrete idea you already know from Grade 10 — perfect squares — and build from there to expressions with variables and products of radicals.

What you will learn

Identify perfect-square factors inside a radicand and explain why they can be moved outside the radical sign.
Apply the product rule for radicals to break one radical into a product of two radicals and simplify.
Simplify radical expressions involving whole numbers and variable expressions by fully removing all perfect-square factors.
Multiply two radical expressions together and simplify the result using the product rule.

Prerequisite Bridge: Perfect Squares and Square Roots

Before working with the product rule, it helps to recall what a perfect square is. A perfect square is any number (or variable expression) that is the square of a whole number or a simple variable. For example,

4 = 2^2

,

9 = 3^2

,

25 = 5^2

,

36 = 6^2

, and

x^2

is a perfect square because

\sqrt{x^2} = x

(assuming

x \geq 0

). Similarly,

x^4 = (x^2)^2

, so

\sqrt{x^4} = x^2

.

The square root sign

\sqrt{\phantom{0}}

asks: what number multiplied by itself gives the value inside? When the value inside — called the radicand — is a perfect square, the answer is exact and whole. When the radicand is not a perfect square, the root is irrational, and simplifying means pulling any hidden perfect-square factors out to make the expression as clean as possible.

You will also need to remember that

\sqrt{a} \cdot \sqrt{a} = a

for any non-negative value

a

. This is just the definition of a square root, and it becomes a useful clean-up step at the end of many problems.

Perfect squares include $1, 4, 9, 16, 25, 36, 49, 64, 81, 100, \ldots$ and variable powers with even exponents such as $x^2, x^4, x^6$ .
The radicand is the expression written under the radical sign.
$\sqrt{x^2} = x$ when $x \geq 0$ ; more generally, $\sqrt{x^{2n}} = x^n$ for non-negative $x$ .

The Product Rule for Radicals

The core idea of this lesson is one rule: the square root of a product equals the product of the square roots. In symbols, for any non-negative values

a

and

b

, you can write

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

. This rule also works in reverse:

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

.

Why does this work? Think about

\sqrt{36}

. You can calculate it directly as

6

. But

36 = 4 \times 9

, so you could also write

\sqrt{4 \times 9} = \sqrt{4} \cdot \sqrt{9} = 2 \cdot 3 = 6

. Both routes give the same answer, confirming the rule. The rule lets you split one complicated radical into two simpler ones whenever it is helpful to do so.

The strategy for simplifying is to find the largest perfect-square factor of the radicand, split the radical using the product rule, evaluate the perfect-square root, and write the result as a whole number (or expression) multiplied by a simpler radical. A radical is fully simplified when no perfect-square factor remains inside the radical sign other than

1

.

When variables appear under the radical, the same rule applies. For example,

\sqrt{x^5}

can be rewritten as

\sqrt{x^4 \cdot x} = \sqrt{x^4} \cdot \sqrt{x} = x^2\sqrt{x}

. Here,

x^4

is the largest perfect-square factor of

x^5

, so it comes out cleanly and

x

stays inside.

\sqrt{a · b} = \sqrt{a} · \sqrt{b}

Product rule: $\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}$ for $a \geq 0$ , $b \geq 0$ .
Always factor out the largest perfect-square factor to simplify in one step.
A radical is fully simplified when the radicand has no perfect-square factor greater than $1$ .
The rule applies to variable radicands: even exponents come out, odd exponents leave one factor inside.

Simplifying Radicals with Numbers

Here is the step-by-step approach with a numerical example. To simplify

\sqrt{72}

, first list factor pairs of

72

and identify the largest perfect-square factor:

72 = 36 \times 2

, and

36

is a perfect square. Apply the product rule:

\sqrt{72} = \sqrt{36 \times 2} = \sqrt{36} \cdot \sqrt{2} = 6\sqrt{2}

. Since

2

has no perfect-square factor other than

1

, the expression is fully simplified.

A common shortcut trap is choosing a perfect-square factor that is not the largest one. For instance, you might notice that

4

also divides

72

and write

\sqrt{72} = \sqrt{4 \times 18} = 2\sqrt{18}

. This is not wrong — it is just not finished, because

18 = 9 \times 2

still contains a perfect-square factor of

9

. You would need to simplify again:

2\sqrt{18} = 2 \cdot 3\sqrt{2} = 6\sqrt{2}

. Both paths reach the same answer, but picking the largest perfect-square factor first saves a step.

Once you have simplified individual radicals, you can also combine them using the product rule in reverse. To multiply

3\sqrt{5} \times 2\sqrt{7}

, multiply the numbers outside together and the radicands together:

3 \times 2 = 6

and

\sqrt{5} \times \sqrt{7} = \sqrt{35}

. The result is

6\sqrt{35}

. Since

35 = 5 \times 7

has no perfect-square factor, this is fully simplified.

Find the largest perfect-square factor of the radicand before splitting.
If you use a non-largest perfect-square factor, you must simplify again — the answer is correct only when no perfect-square factor remains inside.
To multiply expressions like $a\sqrt{m} \times b\sqrt{n}$ , compute $(ab)\sqrt{mn}$ and then check whether $mn$ can be simplified.

Simplifying Radicals with Variables

The product rule extends naturally to variable radicands. The key is to separate even powers (perfect squares) from any leftover odd power. For instance, to simplify

\sqrt{50x^3}

, factor the radicand into its perfect-square part and its leftover part. For the number:

50 = 25 \times 2

. For the variable:

x^3 = x^2 \times x

. So

\sqrt{50x^3} = \sqrt{25 \cdot 2 \cdot x^2 \cdot x} = \sqrt{25} \cdot \sqrt{x^2} \cdot \sqrt{2x} = 5x\sqrt{2x}

.

Notice how the even-power part

x^2

came out as

x

, while the single leftover

x

stayed inside with the

2

. Always group the perfect-square factors together and the non-perfect-square factors together before applying the rule — this keeps the work organized.

When you multiply two radical expressions that contain variables, apply the product rule to combine the radicands, then simplify the result. For example,

\sqrt{6x} \cdot \sqrt{10x^3} = \sqrt{6x \cdot 10x^3} = \sqrt{60x^4}

. Now simplify:

60 = 4 \times 15

and

x^4

is a perfect square, so

\sqrt{60x^4} = \sqrt{4 \cdot 15 \cdot x^4} = 2x^2\sqrt{15}

.

Throughout this lesson, assume all variables represent non-negative values. This ensures that expressions like

\sqrt{x^2} = x

hold without needing to use absolute values, which belong to later study.

\sqrt{a^{2n} · b} = a^n\sqrt{b}

Split $\sqrt{a^{\text{even}} \cdot a^{\text{remainder}}}$ : the even part exits the radical, the remainder stays inside.
Group all perfect-square factors together before applying the product rule.
After multiplying two radicals, always check whether the new radicand can be simplified.
Assume all variables are non-negative throughout this course.

Putting It All Together: A Strategy Summary

When you encounter a radical expression to simplify, use this consistent four-step approach. First, factor the radicand completely, separating perfect-square factors from non-perfect-square factors. Second, apply the product rule to write the radical as a product of two radicals — one containing only perfect squares and one containing only non-perfect-square factors. Third, evaluate the perfect-square radical. Fourth, write the coefficient (number or variable) in front of the remaining radical and confirm that the radicand is fully simplified.

When multiplying two radical expressions, an extra step comes first: combine the radicands using the product rule, then follow the four steps above on the resulting single radical.

Simplification matters because it allows you to compare and combine radical expressions clearly. Two expressions that look different, such as

\sqrt{200}

and

10\sqrt{2}

, are actually equal, and simplifying reveals this. Throughout MCR3U, you will see simplified radical form appear in exact answers to quadratic equations, in trigonometric exact values like

\frac{\sqrt{3}}{2}

, and in other contexts where decimal approximations are not precise enough.

Four steps: factor the radicand → split with the product rule → evaluate the perfect-square root → confirm full simplification.
For products of radicals, combine radicands first, then simplify.
Simplified radical form gives exact, unambiguous answers — decimals round and lose precision.

Common Radicands and Their Simplified Forms

Radicand inside $\sqrt{\phantom{0}}$	Largest Perfect-Square Factor	Simplified Form
$12$	$4$	$2\sqrt{3}$
$45$	$9$	$3\sqrt{5}$
$72$	$36$	$6\sqrt{2}$
$x^5$	$x^4$	$x^2\sqrt{x}$
$50x^3$	$25x^2$	$5x\sqrt{2x}$
$98x^6$	$49x^6$	$7x^3\sqrt{2}$

Worked example

Simplifying a Numerical Radical Expression

Simplify

4\sqrt{180}

.

Factor the radicand
Write $180$ as a product that includes the largest perfect-square factor you can find. List factors of $180$ : $180 = 4 \times 45 = 9 \times 20 = 36 \times 5$ . The largest perfect-square factor is $36$ , so write $180 = 36 \times 5$ .
$180 = 36 × 5$
Apply the product rule
Use $\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}$ to split the radical into the perfect-square part and the leftover part.
$\sqrt{180} = \sqrt{36 × 5} = \sqrt{36} · \sqrt{5}$
Evaluate the perfect-square root
Compute $\sqrt{36} = 6$ because $6^2 = 36$ .
$\sqrt{36} · \sqrt{5} = 6\sqrt{5}$
Multiply by the outside coefficient
The original expression had a coefficient of $4$ in front of the radical. Multiply $4$ by the $6$ that came out of the radical.
$4 × 6\sqrt{5} = 24\sqrt{5}$
Confirm full simplification
Check that $5$ has no perfect-square factor other than $1$ . Since $5$ is prime, no further simplification is possible.

Answer:

4\sqrt{180} = 24\sqrt{5}

Check: Verify numerically:

\sqrt{180} \approx 13.416

, so

4\sqrt{180} \approx 53.67

. Also,

24\sqrt{5} \approx 24 \times 2.236 \approx 53.67

. The values match, confirming the answer.

Worked example

Multiplying and Simplifying Radical Expressions with Variables

Simplify

3\sqrt{2x^3} \times 5\sqrt{8x}

, where

x \geq 0

.

Multiply the coefficients
Multiply the numbers in front of the radicals together. Here $3 \times 5 = 15$ .
$3 × 5 = 15$
Combine the radicands using the product rule
Use the product rule in reverse to combine the two radicals into one: $\sqrt{2x^3} \cdot \sqrt{8x} = \sqrt{2x^3 \cdot 8x}$ . Multiply inside: $2 \times 8 = 16$ and $x^3 \times x = x^4$ .
$\sqrt{2x^3} · \sqrt{8x} = \sqrt{16x^4}$
Identify perfect-square factors in the new radicand
Inside $\sqrt{16x^4}$ , both $16$ and $x^4$ are perfect squares: $16 = 4^2$ and $x^4 = (x^2)^2$ . The entire radicand is a perfect square.
$\sqrt{16x^4} = \sqrt{(4x^2)^2}$
Evaluate the radical
Since the radicand is a perfect square, the radical evaluates to exactly $4x^2$ .
$\sqrt{(4x^2)^2} = 4x^2$
Write the final answer
Multiply the coefficient found in the first step, $15$ , by the result from the radical, $4x^2$ .
$15 × 4x^2 = 60x^2$

Answer:

3\sqrt{2x^3} \times 5\sqrt{8x} = 60x^2

Check: Let

x = 4

. Left side:

3\sqrt{2(64)} \times 5\sqrt{8(4)} = 3\sqrt{128} \times 5\sqrt{32} = 3 \times 8\sqrt{2} \times 5 \times 4\sqrt{2} = 15 \times 32 \times (\sqrt{2})^2 = 480 \times 2 = 960

. Right side:

60(4^2) = 60 \times 16 = 960

. Both sides equal

960

, confirming the answer.

Common mistakes and how to avoid them

Choosing a perfect-square factor that is not the largest one, for example writing

\sqrt{72} = \sqrt{4 \times 18} = 2\sqrt{18}

and stopping there.

Correction: Always find the largest perfect-square factor. Here,

36

is the largest, giving

\sqrt{72} = 6\sqrt{2}

in one step. If you use a smaller factor, keep simplifying until no perfect-square factor remains inside.

Treating

\sqrt{a + b}

as

\sqrt{a} + \sqrt{b}

, for example writing

\sqrt{9 + 16} = 3 + 4 = 7

.

Correction: The product rule applies to multiplication only, not addition or subtraction.

\sqrt{9 + 16} = \sqrt{25} = 5

, not

7

. Always evaluate or simplify what is inside the radical before splitting.

Forgetting to multiply the outside coefficient by the number that exits the radical, for example simplifying

4\sqrt{45}

as

4\sqrt{9 \times 5} = 4 \times \sqrt{5} = 4\sqrt{5}

instead of

12\sqrt{5}

.

Correction: When a number exits the radical, it becomes a factor of the coefficient outside. Here

\sqrt{9} = 3

, so

4\sqrt{45} = 4 \times 3\sqrt{5} = 12\sqrt{5}

.

Leaving an odd-power variable entirely inside the radical, for example writing

\sqrt{x^5}

as

\sqrt{x^5}

without simplifying.

Correction: Split the odd power into the largest even power plus one leftover:

x^5 = x^4 \cdot x

. Then

\sqrt{x^5} = x^2\sqrt{x}

.

Stopping simplification after combining radicands when multiplying, without checking whether the new radicand can be simplified further.

Correction: After applying the product rule to combine two radicals, always inspect the resulting radicand for perfect-square factors and simplify if any remain.

Lesson summary

The product rule states that $\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}$ for non-negative values $a$ and $b$ , and it works in both directions.
To simplify a radical, find the largest perfect-square factor of the radicand, apply the product rule to split it, evaluate the perfect-square root, and place the result as a coefficient outside the radical.
A radical expression is fully simplified when the radicand contains no perfect-square factor greater than $1$ .
For variable radicands, separate even-power factors (which exit the radical) from odd-power leftovers (which stay inside).
To multiply two radical expressions, multiply coefficients together and radicands together using the product rule, then simplify the resulting radical.
Simplified radical form gives exact answers and makes it easy to identify when two expressions are equal.

Check your understanding

Question 1

Which of the following is the fully simplified form of

\sqrt{48}

?

$2\sqrt{12}$
$4\sqrt{3}$
$6\sqrt{2}$
$3\sqrt{16}$

Show answer and explanation

4\sqrt{3}

48 = 16 \times 3

, and

16

is the largest perfect-square factor. So

\sqrt{48} = \sqrt{16} \cdot \sqrt{3} = 4\sqrt{3}

. Option A is not fully simplified because

\sqrt{12}

can be simplified further. Option C is incorrect (

6\sqrt{2} = \sqrt{72} \neq \sqrt{48}

). Option D is not standard simplified form since

\sqrt{16} = 4

should be evaluated.

Question 2

Simplify

\sqrt{x^7}

, where

x \geq 0

.

$x^3\sqrt{x}$
$x^2\sqrt{x^3}$
$7\sqrt{x}$
$x^6\sqrt{x}$

Show answer and explanation

x^3\sqrt{x}

Split the exponent into the largest even part plus the remainder:

x^7 = x^6 \cdot x

. Then

\sqrt{x^7} = \sqrt{x^6 \cdot x} = \sqrt{x^6} \cdot \sqrt{x} = x^3\sqrt{x}

. Option B leaves

\sqrt{x^3}

unsimplified. Option C incorrectly treats the exponent as a coefficient. Option D is too large;

\sqrt{x^{12}} = x^6

, not

\sqrt{x^6}

.

Question 3

What is the simplified form of

2\sqrt{3} \times 4\sqrt{15}

?

$8\sqrt{18}$
$24\sqrt{5}$
$8\sqrt{45}$
$6\sqrt{45}$

Show answer and explanation

24\sqrt{5}

Multiply the coefficients:

2 \times 4 = 8

. Combine the radicands:

\sqrt{3} \cdot \sqrt{15} = \sqrt{45}

. So far:

8\sqrt{45}

. Now simplify

\sqrt{45}

:

45 = 9 \times 5

, so

\sqrt{45} = 3\sqrt{5}

. Final answer:

8 \times 3\sqrt{5} = 24\sqrt{5}

. Option C shows the unsimplified intermediate step, and Options A and D use incorrect radicand multiplication.

Question 4

A student writes

\sqrt{25 + 144} = 5 + 12 = 17

. What is the correct value?

$17$
$13$
$\sqrt{17}$
$\sqrt{119}$

Show answer and explanation

13

The product rule applies to multiplication, not addition. You must simplify inside the radical first:

25 + 144 = 169

. Then

\sqrt{169} = 13

because

13^2 = 169

. The student incorrectly split a sum as though it were a product.

Key terms

Radical: An expression that uses a root symbol $\sqrt{\phantom{0}}$ . In this course, radicals refer to square roots unless otherwise stated.
Radicand: The number or expression written under the radical sign. For example, in $\sqrt{72}$ , the radicand is $72$ .
Perfect square: A number or variable expression that is the square of a whole number or a simple variable expression. Examples: $36 = 6^2$ and $x^4 = (x^2)^2$ .
Product rule for radicals: The rule that $\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}$ for non-negative values $a$ and $b$ . It allows a single radical to be split into a product of two radicals, or two radicals to be combined into one.
Simplified radical form: A radical expression is in simplified form when its radicand contains no perfect-square factor greater than $1$ and all perfect-square factors have been moved outside the radical.
Coefficient: The number (or expression) multiplied by a radical from the outside. In $5\sqrt{3}$ , the coefficient is $5$ .

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About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Ontario Grade 11 Mathematics (MCR3U), expectation A3.2. It is a study resource, not an official curriculum publication.

Before publication, content is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability. Errors can still occur, so corrections are welcomed.

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