Radical Rules: All 7 Properties of nth Roots (Cheat Sheet)

Q: Can you add or subtract radicals like you multiply them?

No. There is no product-style rule for addition: $\sqrt{a + b} \neq \sqrt{a} + \sqrt{b}$. You can only add or subtract radicals when they share the same index and the same radicand (like terms). For example, $3\sqrt{5} + 2\sqrt{5} = 5\sqrt{5}$, but $\sqrt{2} + \sqrt{3}$ cannot be simplified further.

Q: When do you need to use absolute value with radicals?

When the index $n$ is even, $\sqrt[n]{a^n} = |a|$, not simply $a$. This is because an even root always returns a nonnegative result. For example, $\sqrt{(-3)^2} = \sqrt{9} = 3 = |-3|$. When $n$ is odd, absolute value is not needed because odd roots preserve the sign of the input.

Radical Rules
Root Rules
nth Root Rules

Algebra rules for nth roots are listed below. Radical expressions can be rewritten using exponents, so the rules below are a subset of the exponent rules.

For all of the following, n is an integer and n ≥ 2.
Definitions 1. if both b ≥ 0 and bⁿ = a.	Examples because 2³ = 8.
2. If n is odd then .
3. If n is even then .
4. If a ≥ 0 then .	and
Distributing (a ≥ 0 and b ≥ 0) 1. 2. (b ≠ 0)	Examples
3. ( multiplied by itself n times equals a)
4. (m ≥ 0)
Rationalizing the Denominator (a > 0, b > 0, c > 0)	Examples


Example
Careful!! 1. 2. 3.	Examples

Key Formula

\sqrt[n]{a} = a^{\tfrac{1}{n}}

\sqrt[n]{a \cdot b} = \sqrt[n]{a}\;\cdot\;\sqrt[n]{b}

\sqrt[n]{\frac{a}{b}} = \frac{\sqrt[n]{a}}{\sqrt[n]{b}},\quad b \neq 0

\left(\sqrt[n]{a}\right)^n = a

\sqrt[n]{a^m} = a^{\tfrac{m}{n}}

Where:

$a, b$ = Nonnegative real numbers (for even n) or any real numbers (for odd n)
$n$ = The index of the radical; an integer with n ≥ 2
$m$ = An integer exponent applied to the radicand

Worked Example

Problem: Simplify

\sqrt{50} \cdot \sqrt{8}

Step 1: Use the product rule for radicals to combine the two square roots into one.

\sqrt{50} \cdot \sqrt{8} = \sqrt{50 \cdot 8} = \sqrt{400}

Step 2: Evaluate the square root of 400.

\sqrt{400} = 20

Step 3: Alternatively, simplify each radical first:

\sqrt{50} = 5\sqrt{2}

and

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

5\sqrt{2} \cdot 2\sqrt{2} = 10 \cdot (\sqrt{2})^2 = 10 \cdot 2 = 20

Answer:

\sqrt{50} \cdot \sqrt{8} = 20

Another Example

This example involves a cube root instead of a square root and demonstrates rationalizing a denominator that contains an nth root, a common application of radical rules.

Problem: Rationalize the denominator of

\dfrac{6}{\sqrt[3]{4}}

Step 1: Rewrite the denominator using exponents:

\sqrt[3]{4} = 4^{1/3} = 2^{2/3}

. To clear the radical, you need the exponent to become a whole number. Multiply numerator and denominator by

\sqrt[3]{2}

(i.e.,

2^{1/3}

\frac{6}{\sqrt[3]{4}} \cdot \frac{\sqrt[3]{2}}{\sqrt[3]{2}}

Step 2: In the denominator, apply the product rule for radicals.

\sqrt[3]{4} \cdot \sqrt[3]{2} = \sqrt[3]{8} = 2

Step 3: Write the simplified fraction.

\frac{6\,\sqrt[3]{2}}{2} = 3\sqrt[3]{2}

Answer:

\dfrac{6}{\sqrt[3]{4}} = 3\sqrt[3]{2}

Frequently Asked Questions

What is the difference between radical rules and exponent rules?

Radical rules are a direct consequence of exponent rules. Since

\sqrt[n]{a} = a^{1/n}

, every radical identity can be derived from a corresponding exponent identity. Exponent rules cover all rational and integer powers, while radical rules specifically address expressions written with the radical symbol

\sqrt[n]{\phantom{x}}

Can you add or subtract radicals like you multiply them?

No. There is no product-style rule for addition:

\sqrt{a + b} \neq \sqrt{a} + \sqrt{b}

. You can only add or subtract radicals when they share the same index and the same radicand (like terms). For example,

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

, but

\sqrt{2} + \sqrt{3}

cannot be simplified further.

When do you need to use absolute value with radicals?

When the index

n

is even,

\sqrt[n]{a^n} = |a|

, not simply

a

. This is because an even root always returns a nonnegative result. For example,

\sqrt{(-3)^2} = \sqrt{9} = 3 = |-3|

. When

n

is odd, absolute value is not needed because odd roots preserve the sign of the input.

Radical Form vs. Exponential Form

	Radical Form	Exponential Form
Notation	$\sqrt[n]{a}$	$a^{1/n}$
Product rule	$\sqrt[n]{ab} = \sqrt[n]{a}\cdot\sqrt[n]{b}$	$(ab)^{1/n} = a^{1/n}\cdot b^{1/n}$
Quotient rule	$\sqrt[n]{a/b} = \sqrt[n]{a}\,/\,\sqrt[n]{b}$	$(a/b)^{1/n} = a^{1/n}\,/\, b^{1/n}$
Power of a radical	$\sqrt[n]{a^m} = (\sqrt[n]{a})^m$	$a^{m/n}$
Best used when	Simplifying or rationalizing denominators	Performing algebra with fractional exponents

Why It Matters

Radical rules appear throughout algebra, geometry, and precalculus—from simplifying square roots in the Pythagorean theorem to solving radical equations and working with rational exponents. Standardized tests like the SAT and ACT regularly ask you to simplify radical expressions or rationalize denominators. Mastering these rules also prepares you for calculus, where rewriting radicals as fractional exponents is essential for differentiation and integration.

Common Mistakes

Mistake: Assuming

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

Correction: The product rule lets you split radicals over multiplication, not addition. For instance,

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

, but

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

. These are not equal.

Mistake: Forgetting absolute value when the index is even: writing

\sqrt{x^2} = x

instead of

\sqrt{x^2} = |x|

Correction: An even-index radical always returns a nonnegative value. If

x

could be negative, you must write

\sqrt{x^2} = |x|

. This ensures the result is correct for all real values of

x

Related Terms

nth Root — The core operation that radical rules describe
Radical — The symbol and expression these rules apply to
Exponent Rules — Parent set of rules from which radical rules derive
Exponent — Radicals rewrite as fractional exponents
Rationalizing the Denominator — Key application of radical multiplication rules
Square Root Rules — Special case of radical rules where n = 2
Absolute Value Rules — Needed when simplifying even-index radicals
Distributing Rules — Used when expanding products involving radicals