Simplify and operate on square-root expressions

Alberta Grade 11 Mathematics

Number and Logic

Recognize perfect-square factors, combine like radicals, and use square-root rules carefully

A square root asks which non-negative number, multiplied by itself, gives the number inside the root. For example, since

5^2=25

, the principal square root of

25

is

5

. The word principal means the non-negative square root. In this lesson, you will use perfect squares to simplify roots, then apply the usual rules for adding, subtracting, multiplying, and dividing expressions. Keep the root sign over the entire intended radicand, the number or expression inside the root.

What you will learn

Simplify a square root by taking out perfect-square factors.
Add and subtract square-root terms only when their radical parts match.
Multiply and divide square-root expressions using square-root rules.
Check that a simplified expression has the same value as the original.

1. Review: perfect squares and factors

A perfect square is a number made by multiplying an integer by itself. Some useful perfect squares are CAD 1, 4, 9, 16, 25, 36, 49, 64, 81, and

100

. Knowing these helps you spot factors that can be taken out of a square root.

For example,

72

is not a perfect square, but it has the perfect-square factor

36

. Since

72=36\times2

, its square root can be written as

\sqrt{36}\sqrt{2}

. The first root is

6

, so

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

.

A square root of a real number is defined here when the number under the root is zero or positive. Also, a denominator cannot equal zero. These restrictions matter when expressions include variables.

\sqrt{ab}=\sqrt{a}\sqrt{b}\quad(a\geq0,\ b\geq0)

Look for the largest perfect-square factor of the radicand.
For a non-negative number $a$ , $\sqrt{a^2}=a$ .
The principal square root is non-negative.

2. Simplify roots, then combine like radicals

To simplify a root, separate out a perfect-square factor. For example,

\sqrt{45}=\sqrt{9\times5}=3\sqrt{5}

. The number outside the root is called the coefficient. The remaining root part is called the radical part.

Like radicals have the same simplified radical part, just as like terms have the same variable part. You can add or subtract their coefficients. For instance,

2\sqrt{7}+5\sqrt{7}=7\sqrt{7}

. But

\sqrt{7}

and

\sqrt{5}

are not like radicals, so they cannot be combined by addition.

Always simplify each root before deciding whether terms are alike. In

\sqrt{12}+\sqrt{27}

, the simplified forms are

2\sqrt{3}

and

3\sqrt{3}

. They are now like radicals and combine to

5\sqrt{3}

.

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

Simplify each radical before adding or subtracting.
Combine coefficients only when the radical parts match.
Do not add the numbers inside unlike roots.

3. Multiply and divide square-root expressions

When multiplying roots, multiply the radicands. You can also multiply the coefficients outside the roots. Then simplify the result if it has a perfect-square factor. For example,

\sqrt{6}\times\sqrt{24}=\sqrt{144}=12

.

When dividing roots, divide the radicands if the divisor is non-zero and the quotient is defined. Simplifying first can make the arithmetic easier. For example,

\sqrt{75}/\sqrt{3}=\sqrt{25}=5

.

A frequent error is to apply the multiplication rule to addition. In general,

\sqrt{a}+\sqrt{b}

is not equal to

\sqrt{a+b}

. Check with

a=9

and

b=16

: the left side is

3+4=7

, while the right side is

5

. The product and quotient rules do not create corresponding addition and subtraction rules.

\sqrt{a}\sqrt{b}=\sqrt{ab}\quad(a,b\geq0)

Multiply or divide the radicands, then simplify.
For real square roots in a quotient, the divisor must be non-zero.
There is no general rule that combines addition or subtraction inside one root.

4. A reliable checking method

A simplified expression should have the same value as the original. One check is to square a simple result. Since

6\sqrt{2}

squared is

36\times2=72

, it matches

\sqrt{72}

, and both values are non-negative.

For sums, check that the radical parts really match before combining coefficients. For products and quotients, check the arithmetic in the radicand and simplify any perfect-square factor. With variables, also check that the original roots are defined and that no denominator is zero.

Try these checks as you work: simplify

\sqrt{80}

; combine

3\sqrt{2}+\sqrt{18}

; and find

\sqrt{8}\times\sqrt{2}

. The worked examples show each kind of reasoning.

Check equivalence, not just whether the expression looks shorter.
The final radical should not contain a perfect-square factor greater than $1$ .
Keep any restrictions from the original expression.

Worked example

Simplify a square root

Simplify

\sqrt{108}

.

Find a square factor
The largest perfect-square factor of $108$ is $36$ , because $108=36\times3$ . Separating this factor lets us take its square root outside.
$\sqrt{108}=\sqrt{36}\sqrt{3}$
Take the square root
Since $\sqrt{36}=6$ , the simplified expression is $6\sqrt{3}$ . The remaining radicand, $3$ , has no perfect-square factor greater than $1$ .
$\sqrt{108}=6\sqrt{3}$

Answer:

6\sqrt{3}

Check: Squaring the non-negative result gives

(6\sqrt{3})^2=36\times3=108

, which matches the original radicand.

Worked example

Add and subtract after simplifying

Simplify

2\sqrt{12}+\sqrt{27}-\sqrt{3}

.

Simplify each root
The roots $\sqrt{12}$ and $\sqrt{27}$ each contain a perfect-square factor. Rewrite them before combining, while keeping the coefficients and subtraction sign.
$2\sqrt{12}=4\sqrt{3},\quad\sqrt{27}=3\sqrt{3}$
Combine like radicals
All three terms now have the radical part $\sqrt{3}$ . Add and subtract their coefficients: $4+3-1=6$ .
$4\sqrt{3}+3\sqrt{3}-\sqrt{3}=6\sqrt{3}$

Answer:

6\sqrt{3}

Check: Each original term is a multiple of

\sqrt{3}

: the coefficients are

4

,

3

, and

-1

. Their total is

6

, so the result is consistent.

Worked example

Multiply and simplify

Simplify

3\sqrt{5}\times2\sqrt{15}

.

Multiply coefficients and roots
Multiply the outside coefficients together and multiply the radicands together. Both radicands are positive, so the product rule applies.
$3\times2=6,\quad\sqrt{5}\sqrt{15}=\sqrt{75}$
Simplify the product
Since $75=25\times3$ , take $\sqrt{25}=5$ outside the root. Then multiply that factor by the coefficient $6$ .
$6\sqrt{75}=6\times5\sqrt{3}=30\sqrt{3}$

Answer:

30\sqrt{3}

Check: The product before simplification is

6\sqrt{75}

. Squaring the radical part of the answer gives

(30\sqrt{3})^2=2700

, and

(6\sqrt{75})^2=36\times75=2700

; both are positive.

Common mistakes and how to avoid them

Writing

\sqrt{12}=6\sqrt{2}

.

Correction: Use a product that equals the radicand:

12=4\times3

, so

\sqrt{12}=2\sqrt{3}

.

Combining unlike radicals, such as

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

.

Correction: Addition does not combine radicands. Combine terms only after simplification shows matching radical parts.

Forgetting to multiply the coefficients outside the roots.

Correction: For

a\sqrt{m}\times b\sqrt{n}

, multiply

a

by

b

and multiply the roots, then simplify.

Lesson summary

To simplify a root, take out perfect-square factors.
Simplify each term before adding or subtracting; combine only matching radical parts.
For multiplication and division, operate on the radicands and simplify the result.
Check that the result is equivalent to the original and that all roots and denominators are allowed.

Check your understanding

Question 1

What is the simplified form of

\sqrt{80}

?

$4\sqrt{5}$
$8\sqrt{10}$
$40\sqrt{2}$
$4\sqrt{20}$

Show answer and explanation

4\sqrt{5}

Since

80=16\times5

, its square root is

4\sqrt{5}

. The other options are not equivalent or are not fully simplified.

Question 2

Simplify

\sqrt{18}+2\sqrt{8}

.

$7\sqrt{2}$
$5\sqrt{26}$
$10\sqrt{2}$
$7\sqrt{10}$

Show answer and explanation

7\sqrt{2}

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

and

2\sqrt{8}=4\sqrt{2}

, so the sum is

7\sqrt{2}

.

Question 3

Simplify

\sqrt{6}\times\sqrt{24}

.

$12$
$30$
$\sqrt{30}$
$6\sqrt{4}$

Show answer and explanation

12

\sqrt{6}\times\sqrt{24}=\sqrt{144}=12

.

Key terms

Radicand: The number or expression inside a radical sign.
Perfect square: A number that results from multiplying an integer by itself.
Coefficient: A number multiplying a term, such as the $6$ in $6\sqrt{3}$ .
Like radicals: Square-root terms with the same radical part after simplification.
Principal square root: The non-negative square root of a non-negative number.

Continue through Math 20-2

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About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Alberta Grade 11 Mathematics (Math 20-2), expectation NL3. 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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