Simplifying Square Roots

Learn how to simplify square roots by finding and extracting perfect square factors.

Intermediate25 minLesson

Definition

To simplify a square root means to rewrite it in its simplest form by extracting perfect square factors.

A square root is in simplest form when:

The number under the radical has no perfect square factors other than 1
There are no fractions under the radical
There are no radicals in the denominator

Key Property:

\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}

We use this to separate perfect squares:

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

Try it now

What is

\sqrt{36}

Worked Examples

Simplify $\sqrt{72}$

List perfect squares less than 72

$1, 4, 9, 16, 25, 36, 49, 64$ → Perfect squares to check

Find which divide 72 evenly

$72 \div 4 = 18$ \checkmark, $72 \div 9 = 8$ \checkmark, $72 \div 36 = 2$ \checkmark → 4, 9, and 36 are factors

Choose the largest: 36

$\sqrt{72} = \sqrt{36 \times 2}$ → Factor out 36

Apply the product rule

$\sqrt{36 \times 2} = \sqrt{36} \times \sqrt{2}$ → Separate the roots

Simplify the perfect square

$\sqrt{36} = 6$ → $6\sqrt{2}$

Answer: $\sqrt{72} = 6\sqrt{2}$

Common Mistakes

Stopping too early: writing $\sqrt{72} = 2\sqrt{18}$ instead of $6\sqrt{2}$

Why it's wrong: 18 still has a perfect square factor (9). Always check if the remaining radicand can be simplified further.

Correct: Continue simplifying: $2\sqrt{18} = 2\sqrt{9 \times 2} = 2 \times 3\sqrt{2} = 6\sqrt{2}$

Writing $\sqrt{50} = 25\sqrt{2}$ instead of $5\sqrt{2}$

Why it's wrong: We take the SQUARE ROOT of 25, not 25 itself. $\sqrt{25} = 5$ , not 25.

Correct: $\sqrt{50} = \sqrt{25 \times 2} = \sqrt{25} \times \sqrt{2} = 5\sqrt{2}$

Thinking $\sqrt{a + b} = \sqrt{a} + \sqrt{b}$

Why it's wrong: Square roots do NOT distribute over addition! Only over multiplication.

Correct: $\sqrt{9 + 16} = \sqrt{25} = 5$ , but $\sqrt{9} + \sqrt{16} = 3 + 4 = 7$ . These are not equal!

Interactive Visual

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Practice Problems

16 problems

Problem 1 of 16

Easy

What is $\sqrt{36}$ ?

Why It Matters

Simplifying square roots is essential in mathematics because:

Exact answers: $6\sqrt{2}$ is exact, while $8.485...$ is an approximation
Easier calculations: Simplified forms make further operations much simpler
Recognizing patterns: Seeing $\sqrt{50} = 5\sqrt{2}$ helps compare values
Real applications: Used in geometry (diagonal of a square), physics (velocity formulas), and engineering

Simplified radicals are the standard way to express irrational numbers in mathematics!

Real World Applications

Diagonal of a Square

Finding the diagonal of a square uses the Pythagorean theorem and requires simplifying square roots.

Example:

A square has side length 6 cm. Its diagonal is $\sqrt{6^2 + 6^2} = \sqrt{72} = 6\sqrt{2}$ cm.

1Try It Yourself

A square garden has sides of 10 meters.

What is the exact length of the diagonal path across it?

Step 1: Write the mathematical expression

Use the Pythagorean theorem: $\sqrt{10^2 + 10^2}$

Screen Sizes

TV and monitor sizes are measured diagonally, which involves square roots.

Example:

A monitor is 40 cm wide and 30 cm tall. The diagonal is $\sqrt{40^2 + 30^2} = \sqrt{2500} = 50$ cm.

2Try It Yourself

A tablet screen is 24 cm wide and 18 cm tall.

What is the diagonal measurement?

Step 1: Write the mathematical expression

Calculate $\sqrt{24^2 + 18^2}$

Key Takeaways

1To simplify $\sqrt{n}$ , find the largest perfect square factor of $n$
2Use the product rule: $\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}$
3Prime factorization helps find perfect square factors (pairs of primes)
4A square root is simplified when no perfect square factors remain under the radical
5Always check if your answer can be simplified further

Frequently Asked Questions

Check the number under the radical. If its only perfect square factor is 1 (no repeated prime factors), it is fully simplified. For example,

\sqrt{30}

is simplified because

30 = 2 \times 3 \times 5

has no repeated primes.

Glossary

Radicand: The number under the radical sign. In $\sqrt{72}$ , the radicand is 72.
Perfect square: A number that is the square of an integer: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, ...
Simplest form: A square root where the radicand has no perfect square factors other than 1.
Product rule for radicals: $\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}$ for non-negative $a$ and $b$ .

Formula Card

Product Rule

\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}

Separate a square root into the product of two square roots

Simplification Pattern

\sqrt{n^2 \times m} = n\sqrt{m}

Extract the perfect square factor from under the radical

Perfect Squares

1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144

Memorize these perfect squares up to 144