Can I simplify \sqrt{24} in a different way?

Yes, as long as you factor 24 differently, you'll eventually reach the same simplified form. For example, 24 = 4 \times 6 gives \sqrt{24} = \sqrt{4 \cdot 6} = 2\sqrt{6}, and 24 = 2 \times 12 = 2 \times 4 \times 3 also gives \sqrt{24} = 2\sqrt{6}. The key is finding the largest perfect square factor to simplify most efficiently in one step.

Why do a and b have to be non-negative?

In the real numbers, you cannot take the square root of a negative number—it is undefined. If a or b were negative, then the product ab under the radical might be negative, making \sqrt{ab} undefined. Requiring a, b \ge 0 ensures the square root exists and is a real number.

How is this method related to simplifying fractions?

Both use similar ideas: you're extracting common factors to write the expression in a cleaner form. With fractions like \frac{12}{18}, you find the GCF (6) and cancel to get \frac{2}{3}. With radicals like \sqrt{12}, you find the largest perfect square factor (4) and extract it outside the radical to get 2\sqrt{3}. The method is parallel but adapted to radicals.

What if a number under the radical doesn't have a perfect square factor?

Then it is already fully simplified. For example, \sqrt{7} has no perfect square factors other than 1, so it cannot be simplified further. Similarly, \sqrt{11}, \sqrt{13}, and other radicals with prime numbers under them are already in simplest form and no simplification is possible.

Simplifying Radicals

Break down square roots of products into simpler parts you can work with—essential for simplifying radical expressions in algebra.

\sqrt{ab} = \sqrt{a}\,\sqrt{b}, \quad a, b \ge 0

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What each symbol means

What Simplifying Radicals takes

Symbol	Meaning
$a$	$a$ is a non-negative number (zero or positive) that is one of two factors inside the square root; confusing it with a coefficient outside the radical would lead to incorrect simplifications.
$b$	$b$ is a non-negative number (zero or positive) that is the other factor inside the square root; it must be non-negative because square roots of negative numbers are not real.

When to use it

Reach for this whenever you need to simplify a square root that contains a product.

Level

Usually taught in: Algebra I

Worked examples

1. Simplify a square root with a perfect square factor

Problem

Simplify

\sqrt{12}

$\sqrt{12} = \sqrt{4 \cdot 3}$
We identify that $12 = 4 \times 3$ , where 4 is a perfect square.
$\sqrt{4 \cdot 3} = \sqrt{4}\,\sqrt{3}$
The product rule for radicals lets us separate this into two separate radicals.
$\sqrt{4}\,\sqrt{3} = 2\sqrt{3}$
We evaluate $\sqrt{4} = 2$ since $2^2 = 4$ , leaving $\sqrt{3}$ as is since 3 has no perfect square factors.

Answer: $2\sqrt{3}$

This method works when the radicand has a perfect square factor: we extract it outside the radical and leave any remaining factors inside. This gives us the simplest form of the radical.

2. Multiply two radicals and simplify the result

Problem

Simplify

\sqrt{5} \cdot \sqrt{20}

$\sqrt{5} \cdot \sqrt{20} = \sqrt{5 \cdot 20}$
Using the product rule in reverse, we combine two radicals into one radical with their product inside.
$\sqrt{5 \cdot 20} = \sqrt{100}$
We multiply the numbers: $5 \times 20 = 100$ .
$\sqrt{100} = 10$
Since $10^2 = 100$ , the square root of 100 is 10, which is a whole number.

Answer: $10$

When you multiply two radicals, combining them first using the product rule in reverse can yield a perfect square that simplifies completely to a whole number, making it easier to see the actual value.

3. Find a game stat multiplier from a simplified radical

Problem

In a video game, your power stat is boosted by a factor of

\sqrt{45}

. Simplify this radical to see what your true stat multiplier is.

$\sqrt{45} = \sqrt{9 \cdot 5}$
We recognize that $9 \times 5 = 45$ and 9 is the largest perfect square that divides 45.
$\sqrt{9 \cdot 5} = \sqrt{9}\,\sqrt{5}$
The product rule for radicals lets us separate the perfect square from the rest.
$\sqrt{9}\,\sqrt{5} = 3\sqrt{5}$
We evaluate $\sqrt{9} = 3$ since $3^2 = 9$ , so the simplified form is $3\sqrt{5}$ (approximately 6.7 for game calculations).

Answer: $3\sqrt{5}$

Simplifying makes the expression clearer and easier to work with in game calculations; instead of $\sqrt{45}$ , you can use the simpler form $3\sqrt{5}$ for any stat calculations or comparisons.

Common mistakes

Where Simplifying Radicals usually goes wrong

The mistake	Why it is wrong	The fix
$\sqrt{9 + 16} = \sqrt{9} + \sqrt{16} = 3 + 4 = 7$	The product rule does not apply to addition; radicals only distribute over multiplication, not over addition.	$\sqrt{9 + 16} = \sqrt{25} = 5$ , not 7. You must add inside the square root first, then simplify.
$\sqrt{12} = \sqrt{4 \cdot 3} = \sqrt{4}\,\sqrt{3}$ and leaving the answer in this form	You have not finished simplifying; $\sqrt{4}\,\sqrt{3}$ is an intermediate form, not the final simplified radical expression.	Continue to the next step: $\sqrt{4}\,\sqrt{3} = 2\sqrt{3}$ .
$2\sqrt{3} = \sqrt{2} \cdot \sqrt{3} = \sqrt{6}$	The coefficient 2 in front is not the same as $\sqrt{2}$ ; the product rule applies to $\sqrt{a} \cdot \sqrt{b}$ , not to a whole number times a radical.	If you want to write $2\sqrt{3}$ under a single radical, rewrite the coefficient: $2\sqrt{3} = \sqrt{4} \cdot \sqrt{3} = \sqrt{4 \cdot 3} = \sqrt{12}$ .

Tips and when to use something else

Look for perfect square factors (4, 9, 16, 25, 36, ...) when deciding how to split the radicand.
Always finish simplifying by evaluating the perfect square root; don't leave $\sqrt{4}\,\sqrt{3}$ as your final answer—complete it to $2\sqrt{3}$ .
This rule applies only to multiplication; if you're adding or subtracting radicals, use the Distributive Property instead to combine like radical terms.
When you have a radical in a denominator, such as $\frac{2}{\sqrt{3}}$ , use Rationalizing the Denominator to clear it after simplifying the radical.

Related formulas

Square RootSquare root finds the non-negative number that, when multiplied by itself, gives your starting number; use it to undo squaring in equations.Prime FactorizationPrime factorization breaks any whole number into a unique multiplication of primes—essential for finding common factors and simplifying fractions.Rationalizing the DenominatorRationalizing the Denominator removes square roots from the bottom of a fraction, making it easier to compare values and perform calculations.

Frequently asked questions

Can I simplify $\sqrt{24}$ in a different way?: Yes, as long as you factor 24 differently, you'll eventually reach the same simplified form. For example, $24 = 4 \times 6$ gives $\sqrt{24} = \sqrt{4 \cdot 6} = 2\sqrt{6}$ , and $24 = 2 \times 12 = 2 \times 4 \times 3$ also gives $\sqrt{24} = 2\sqrt{6}$ . The key is finding the largest perfect square factor to simplify most efficiently in one step.
Why do $a$ and $b$ have to be non-negative?: In the real numbers, you cannot take the square root of a negative number—it is undefined. If $a$ or $b$ were negative, then the product $ab$ under the radical might be negative, making $\sqrt{ab}$ undefined. Requiring $a, b \ge 0$ ensures the square root exists and is a real number.
How is this method related to simplifying fractions?: Both use similar ideas: you're extracting common factors to write the expression in a cleaner form. With fractions like $\frac{12}{18}$ , you find the GCF (6) and cancel to get $\frac{2}{3}$ . With radicals like $\sqrt{12}$ , you find the largest perfect square factor (4) and extract it outside the radical to get $2\sqrt{3}$ . The method is parallel but adapted to radicals.
What if a number under the radical doesn't have a perfect square factor?: Then it is already fully simplified. For example, $\sqrt{7}$ has no perfect square factors other than 1, so it cannot be simplified further. Similarly, $\sqrt{11}$ , $\sqrt{13}$ , and other radicals with prime numbers under them are already in simplest form and no simplification is possible.

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Reviewed 2026-09-18