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Radical Rules

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. b equals the nth root of a if both b ≥ 0 and bn = a.

Examples

The cube root of 8 equals 2, written as ∛8 = 2 because 23 = 8.

2. If n is odd then The nth root of a^n equals a.

The 7th root of (-5)^7 equals -5, illustrating that for odd n, the nth root of x^n = x.

3. If n is even then The nth root of a^n equals the absolute value of a, written as: ⁿ√(aⁿ) = |a|.

The sixth root of (-5)^6 equals the absolute value of -5, which equals 5

4. If a ≥ 0 then The nth root of a^n equals a.

 

The fifth root of π to the fifth power equals π, shown as ⁵√(π⁵) = π   and   The 10th root of π raised to the 10th power equals π

Distributing (a ≥ 0 and b ≥ 0)

1. nth root rule: the nth root of (a times b) equals the nth root of a times the nth root of b

2. The nth root of (a/b) equals the nth root of a divided by the nth root of b.     (b ≠ 0)

Examples

Fourth root of 48 equals fourth root of (16·3) equals fourth root of 16 · fourth root of 3 equals 2·fourth root of 3

Cube root of (1/125) equals cube root of 1 divided by cube root of 125, equals 1/5

3. Nested nth roots equation: the nth root of the nth root of the nth root of a (repeated) equals a   (nth root of a, written as a radical symbol with index n over radicand a multiplied by itself n times equals a)

Nested sixth roots: ⁶√(⁶√(⁶√(⁶√(⁶√(⁶√6))))) = 6

4. nth root rule: the nth root of a^m equals (nth root of a)^m, or a^(m/n)   (m ≥ 0)

 

The fifth root of 2 cubed equals 2^(3/6) equals 2^(1/2) equals the square root of 2
Rationalizing the Denominator
(a > 0, b > 0, c > 0) 		Examples

Rule 1: a divided by nth-root of b equals (a divided by nth-root of b) times (nth-root of b^(n-1) divided by nth-root of...

Simplification: 16/⁴√2 = (16·⁴√8)/2 = 8·⁴√8, achieved by rationalizing with ⁴√(2³) in numerator and denominator.

 
Rule 2: a divided by nth-root of b^m equals a divided by nth-root of b^m, multiplied by nth-root of b^(n-m) over nth-root of...

Simplifying 2/⁵√9: multiply by ⁵√3³/⁵√3³ to get 2⁵√27/3, showing rationalization of a fifth root denominator.

 

Rule 3: a/(b − ⁿ√c) = a(bⁿ⁻¹ + bⁿ⁻²ⁿ√c + … + ⁿ√cⁿ⁻¹) / (bⁿ − c), rationalizing

 

Example

Rationalizing 25/(3−∛2) by multiplying by (3²+3∛2+∛4)/same, yielding 25(9+3∛2+∛4)/25 = 9+3∛2+∛4

 

Careful!!

1. The nth root of (a + b) does not equal the nth root of a plus the nth root of b.

2. The nth root of (a minus b) does not equal the nth root of a minus the nth root of b.

3. The nth root of (a^n + b^n) does not equal a + b

Examples

Cube root of (2+6) ≠ cube root of 2 + cube root of 6; showing radicals do NOT distribute over addition

The fourth root of (6−5) is not equal to the fourth root of 6 minus the fourth root of 5

The 5th root of (2⁵ + 3⁵) ≠ 2 + 3, showing that nth roots do not distribute over addition.

 

See also

nth root, square root rules, distributing rules, absolute value rules, factoring rules

Key Formula

an=a1n\sqrt[n]{a} = a^{\tfrac{1}{n}} a⋅bn=an  ⋅  bn\sqrt[n]{a \cdot b} = \sqrt[n]{a}\;\cdot\;\sqrt[n]{b} abn=anbn,b≠0\sqrt[n]{\frac{a}{b}} = \frac{\sqrt[n]{a}}{\sqrt[n]{b}},\quad b \neq 0 (an)n=a\left(\sqrt[n]{a}\right)^n = a amn=amn\sqrt[n]{a^m} = a^{\tfrac{m}{n}}
Where:
  • a,ba, b = Nonnegative real numbers (for even n) or any real numbers (for odd n)
  • nn = The index of the radical; an integer with n ≥ 2
  • mm = An integer exponent applied to the radicand

Worked Example

Problem: Simplify 50⋅8\sqrt{50} \cdot \sqrt{8}.
Step 1: Use the product rule for radicals to combine the two square roots into one.
50⋅8=50⋅8=400\sqrt{50} \cdot \sqrt{8} = \sqrt{50 \cdot 8} = \sqrt{400}
Step 2: Evaluate the square root of 400.
400=20\sqrt{400} = 20
Step 3: Alternatively, simplify each radical first: 50=52\sqrt{50} = 5\sqrt{2} and 8=22\sqrt{8} = 2\sqrt{2}.
52⋅22=10⋅(2)2=10⋅2=205\sqrt{2} \cdot 2\sqrt{2} = 10 \cdot (\sqrt{2})^2 = 10 \cdot 2 = 20
Answer: 50⋅8=20\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 643\dfrac{6}{\sqrt[3]{4}}.
Step 1: Rewrite the denominator using exponents: 43=41/3=22/3\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 23\sqrt[3]{2} (i.e., 21/32^{1/3}).
643⋅2323\frac{6}{\sqrt[3]{4}} \cdot \frac{\sqrt[3]{2}}{\sqrt[3]{2}}
Step 2: In the denominator, apply the product rule for radicals.
43⋅23=83=2\sqrt[3]{4} \cdot \sqrt[3]{2} = \sqrt[3]{8} = 2
Step 3: Write the simplified fraction.
6 232=323\frac{6\,\sqrt[3]{2}}{2} = 3\sqrt[3]{2}
Answer: 643=323\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 an=a1/n\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 xn\sqrt[n]{\phantom{x}}.
Can you add or subtract radicals like you multiply them?
No. There is no product-style rule for addition: a+b≠a+b\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, 35+25=553\sqrt{5} + 2\sqrt{5} = 5\sqrt{5}, but 2+3\sqrt{2} + \sqrt{3} cannot be simplified further.
When do you need to use absolute value with radicals?
When the index nn is even, ann=∣a∣\sqrt[n]{a^n} = |a|, not simply aa. This is because an even root always returns a nonnegative result. For example, (−3)2=9=3=∣−3∣\sqrt{(-3)^2} = \sqrt{9} = 3 = |-3|. When nn is odd, absolute value is not needed because odd roots preserve the sign of the input.

Radical Form vs. Exponential Form

Radical FormExponential Form
Notationan\sqrt[n]{a}a1/na^{1/n}
Product ruleabn=an⋅bn\sqrt[n]{ab} = \sqrt[n]{a}\cdot\sqrt[n]{b}(ab)1/n=a1/n⋅b1/n(ab)^{1/n} = a^{1/n}\cdot b^{1/n}
Quotient rulea/bn=an / bn\sqrt[n]{a/b} = \sqrt[n]{a}\,/\,\sqrt[n]{b}(a/b)1/n=a1/n / b1/n(a/b)^{1/n} = a^{1/n}\,/\, b^{1/n}
Power of a radicalamn=(an)m\sqrt[n]{a^m} = (\sqrt[n]{a})^mam/na^{m/n}
Best used whenSimplifying or rationalizing denominatorsPerforming 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 a+b=a+b\sqrt{a + b} = \sqrt{a} + \sqrt{b}.
Correction: The product rule lets you split radicals over multiplication, not addition. For instance, 9+16=25=5\sqrt{9 + 16} = \sqrt{25} = 5, but 9+16=3+4=7\sqrt{9} + \sqrt{16} = 3 + 4 = 7. These are not equal.
Mistake: Forgetting absolute value when the index is even: writing x2=x\sqrt{x^2} = x instead of x2=∣x∣\sqrt{x^2} = |x|.
Correction: An even-index radical always returns a nonnegative value. If xx could be negative, you must write x2=∣x∣\sqrt{x^2} = |x|. This ensures the result is correct for all real values of xx.

Related Terms