Vocabulary
Simplest Form of a Radical- A radical with index n is in simplest form if the radicand has no perfect nth powers as factors and any denominator has been rationalized.
Like Radicals- Radical expressions with the same index and radicand.
Like Radicals- Radical expressions with the same index and radicand.
Essential Information
The properties of integer exponents that you have previously learned can also be applied to rational exponents.
Properties of Rational Exponents
Let a and b be real numbers and let m and n be rational numbers. The following properties have the same names as those listed, but now apply to rational exponents as illustrated.
Property:
Properties of Rational Exponents
Let a and b be real numbers and let m and n be rational numbers. The following properties have the same names as those listed, but now apply to rational exponents as illustrated.
Property:
- (a^m) (a^n) = a^m+n
- (a^m)^n = a^mn
- (ab)^m = (a^m)(b^m)
- a^-m = 1/a^m, a does not equal zero
- (a^m)/(a^n) = a^m-n, a does not equal zero
- (a/b)^m = (a^m)/(b^m), b does not equal zero
Properties of Radicals
The third and sixth properties can be expressed using radical notation when m= 1/n for some integer greater than 1.
Product Property
ⁿ√(a x b) = ⁿ√(a) x ⁿ√(b)
Quotient Property of Radicals
ⁿ√(a/b) = ⁿ√(a)/ⁿ√(b), b≠0
The third and sixth properties can be expressed using radical notation when m= 1/n for some integer greater than 1.
Product Property
ⁿ√(a x b) = ⁿ√(a) x ⁿ√(b)
Quotient Property of Radicals
ⁿ√(a/b) = ⁿ√(a)/ⁿ√(b), b≠0
Simplest Form
A radical with index n is in simplest form if the radicand has no perfect nth powers as factors and any denominator has been rationalized.
A radical with index n is in simplest form if the radicand has no perfect nth powers as factors and any denominator has been rationalized.
Like Radicals
Radical expressions with the same index and radicand are like radicals. To add or subtract like radicals, use the distributive property.
Radical expressions with the same index and radicand are like radicals. To add or subtract like radicals, use the distributive property.
Variable Expressions
The properties of rational exponents and radicals can also be applied to expressions involving variables. Because a variable can be positive, negative, or zero, sometimes absolute value is needed when simplifying a variable expression.
When n is odd ⁿ√(xⁿ) = x
When n is even ⁿ√(xⁿ) = lxl
Absolute value is not needed when all variables are assumed to be positive.
The properties of rational exponents and radicals can also be applied to expressions involving variables. Because a variable can be positive, negative, or zero, sometimes absolute value is needed when simplifying a variable expression.
When n is odd ⁿ√(xⁿ) = x
When n is even ⁿ√(xⁿ) = lxl
Absolute value is not needed when all variables are assumed to be positive.