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nth root rules — Properties, Formula & Examples

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=a1/n\sqrt[n]{a} = a^{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=am/n,a≥0\sqrt[n]{a^m} = a^{m/n}, \quad a \geq 0
Where:
  • a,ba, b = Real numbers under the radical (must be non-negative when n is even)
  • nn = The index of the root; an integer where n ≥ 2
  • mm = An integer exponent applied to the radicand

Worked Example

Problem: Simplify 54x53\sqrt[3]{54x^5}, assuming x≥0x \geq 0.
Step 1: Factor the radicand into a perfect cube and a remaining factor.
54x5=27⋅2⋅x3⋅x254x^5 = 27 \cdot 2 \cdot x^3 \cdot x^2
Step 2: Apply the product rule for nth roots: split the radical over multiplication.
27⋅2⋅x3⋅x23=273⋅x33⋅2x23\sqrt[3]{27 \cdot 2 \cdot x^3 \cdot x^2} = \sqrt[3]{27} \cdot \sqrt[3]{x^3} \cdot \sqrt[3]{2x^2}
Step 3: Evaluate the perfect cube roots.
273=3,x33=x\sqrt[3]{27} = 3, \quad \sqrt[3]{x^3} = x
Step 4: Combine the simplified factors.
3x 2x233x\,\sqrt[3]{2x^2}
Answer: 54x53=3x 2x23\sqrt[3]{54x^5} = 3x\,\sqrt[3]{2x^2}

Another Example

This example focuses on rationalizing the denominator with a fourth root, whereas the first example used the product rule to simplify a cube root expression.

Problem: Rationalize the denominator of 584\dfrac{5}{\sqrt[4]{8}}.
Step 1: Rewrite the denominator using prime factorization.
84=234\sqrt[4]{8} = \sqrt[4]{2^3}
Step 2: To make the radicand a perfect fourth power, multiply by the factor that raises the exponent to 4. You need one more factor of 2 inside the radical.
5234⋅2424=5 24244\frac{5}{\sqrt[4]{2^3}} \cdot \frac{\sqrt[4]{2}}{\sqrt[4]{2}} = \frac{5\,\sqrt[4]{2}}{\sqrt[4]{2^4}}
Step 3: Simplify the denominator. Since 24=162^4 = 16, the fourth root is exactly 2.
244=2\sqrt[4]{2^4} = 2
Step 4: Write the final simplified expression.
5 242\frac{5\,\sqrt[4]{2}}{2}
Answer: 584=5 242\dfrac{5}{\sqrt[4]{8}} = \dfrac{5\,\sqrt[4]{2}}{2}

Frequently Asked Questions

What is the difference between nth root rules and exponent rules?
Every nth root can be written as a fractional exponent: an=a1/n\sqrt[n]{a} = a^{1/n}. Because of this, nth root rules are really a specific subset of exponent rules. The product rule abn=an⋅bn\sqrt[n]{ab} = \sqrt[n]{a}\cdot\sqrt[n]{b} is equivalent to (ab)1/n=a1/n⋅b1/n(ab)^{1/n} = a^{1/n} \cdot b^{1/n}. If you know your exponent rules, you already know the nth root rules — they are just written in radical notation.
Can you take an nth root of a negative number?
It depends on whether the index nn is odd or even. When nn is odd, the nth root of a negative number is negative; for example, −83=−2\sqrt[3]{-8} = -2. When nn is even, the nth root of a negative number is not a real number because no real number raised to an even power gives a negative result.
When do you need absolute value with nth roots?
Absolute value is needed when nn is even and the variable could be negative. For example, x44=∣x∣\sqrt[4]{x^4} = |x|, not just xx, because xx might be negative and the even root must return a non-negative result. When nn is odd, you never need absolute value because odd roots preserve the sign.

Nth Root Rules vs. Square Root Rules

Nth Root RulesSquare Root Rules
IndexAny integer n ≥ 2Always n = 2 (index is often omitted)
Product ruleabn=an⋅bn\sqrt[n]{ab} = \sqrt[n]{a}\cdot\sqrt[n]{b}ab=a⋅b\sqrt{ab} = \sqrt{a}\cdot\sqrt{b}
Odd radicandsAllowed when n is odd (e.g., −273=−3\sqrt[3]{-27} = -3)Not allowed (−4\sqrt{-4} is not real)
Exponent forma1/na^{1/n}a1/2a^{1/2}
RationalizingMultiply to create a perfect nth power in denominatorMultiply to create a perfect square in denominator

Why It Matters

You encounter nth root rules throughout Algebra 2, Precalculus, and standardized tests like the SAT and ACT whenever you simplify radical expressions, solve radical equations, or convert between radical and exponential forms. Mastering these rules is essential for working with rational exponents, which appear heavily in calculus when differentiating and integrating power functions. They also show up in applied problems — for instance, computing the geometric mean of nn values or working with formulas in physics and engineering that involve cube roots and fourth roots.

Common Mistakes

Mistake: Distributing the nth root over addition: writing a+bn=an+bn\sqrt[n]{a + b} = \sqrt[n]{a} + \sqrt[n]{b}.
Correction: The product rule lets you split roots over multiplication, not addition. For example, 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 clearly not equal.
Mistake: Forgetting absolute value when the index is even: writing x44=x\sqrt[4]{x^4} = x instead of x44=∣x∣\sqrt[4]{x^4} = |x|.
Correction: When the index is even, the result must be non-negative. If x=−3x = -3, then (−3)44=814=3=∣−3∣\sqrt[4]{(-3)^4} = \sqrt[4]{81} = 3 = |-3|, not −3-3. Always use absolute value unless the problem states that x≥0x \geq 0.

Related Terms