Vocabulary
nth root of a- For an integer n greater than 1, if bⁿ = a, then b is an nth root of a. Written as ⁿ√(a).
Index of a Radical- The integer n, greater than 1, in the expression ⁿ√(a).
Index of a Radical- The integer n, greater than 1, in the expression ⁿ√(a).
Essential Information
There are more types of roots than just square roots and they apply to the same concepts. 2 is a cube root of 8 because 2³ = 8. In general, for an integer n greater than 1, if bⁿ =a, then b is an nth root of a.
nth root of a can be written as a power of a. If you assume the power of a power property applies to rational exponents, then the following is true:
(a⁰⋅⁵)² = a⁽⁰⋅⁵⁾² = a¹ = a
(a⁰⋅³³³)³ = a⁽⁰⋅³³³⁾³ = a¹ = a (a⁰⋅²⁵)⁴ = a⁽⁰⋅²⁵⁾⁴ = a¹ = a and so on...
Real nth Roots of a
Let n be an integer (n>1) and let a be a real number
n is an even integer
a<0 No real nth roots
a=0 One real nth root: ⁿ√(0) = 0
a>0 Two real nth roots: ±ⁿ√(a) = a^1/n
n is an odd integer
a<0 One real nth root: ⁿ√(a) = a^1/n
a=0 One real nth root: ⁿ√(0) = 0
a>0 One real nth root: ⁿ√(a) = a^1/n
nth root of a can be written as a power of a. If you assume the power of a power property applies to rational exponents, then the following is true:
(a⁰⋅⁵)² = a⁽⁰⋅⁵⁾² = a¹ = a
(a⁰⋅³³³)³ = a⁽⁰⋅³³³⁾³ = a¹ = a (a⁰⋅²⁵)⁴ = a⁽⁰⋅²⁵⁾⁴ = a¹ = a and so on...
Real nth Roots of a
Let n be an integer (n>1) and let a be a real number
n is an even integer
a<0 No real nth roots
a=0 One real nth root: ⁿ√(0) = 0
a>0 Two real nth roots: ±ⁿ√(a) = a^1/n
n is an odd integer
a<0 One real nth root: ⁿ√(a) = a^1/n
a=0 One real nth root: ⁿ√(0) = 0
a>0 One real nth root: ⁿ√(a) = a^1/n
Rational Exponents
A rational exponent does not have to be of the form 1/n . Other rational numbers such as 2/3 and -1/2 can also be used as exponents.
Let a^1/n be an nth root of a, and let m be a positive integer.
a^m/n = (a^1/n)^m = (ⁿ√(a))^m
a^-m/n = 1/(a^m/n) = 1/(ⁿ√(a)^m) , a≠0
A rational exponent does not have to be of the form 1/n . Other rational numbers such as 2/3 and -1/2 can also be used as exponents.
Let a^1/n be an nth root of a, and let m be a positive integer.
a^m/n = (a^1/n)^m = (ⁿ√(a))^m
a^-m/n = 1/(a^m/n) = 1/(ⁿ√(a)^m) , a≠0