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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