Download R-2 Exponents and Radicals

R-2 Exponents and Radicals
Definition 1
• a^n , n is an Integer (Z) and a is a Real number
(R)
– a^n = a times a times ...times a (n factors of a)
– a^-n= 1/a^n
– a^0=1
Integer Exponents
Theorem 1 Properties of Integer
Exponents
• For n and m integers and a and b are real
numbers
1. (a^m)(a^n)= a^m+n
2. (a^n)^m= a^mn
3. (ab)^m=(a^m)(b^m)
4. (a/b)^m= (a^m)/(b^m) b cannot = 0
5. (a^m)/(a^n)= a^m-n 1/a^n-m a cannot = 0
Examples
• x^5x^-2
– x^-10 = 1/x^10
– (b^-2)^-4 = b^8
Scientific Notation
• a times 10^n
– n an integer, a in decimal form
– Used in the science field when working with large
numbers
Scientific Notation
Roots of Real Numbers
• Definition of an nth rootFor a natural number n and
a and b real numbers: a is
an nth root of b if a^n=b
Theorem 2: Number of Real nth roots
of a real number b
• Let n be a natural number and let b be a real
number
1. b>0: If n is even, then b has 2 real nth roots,
each the negative of the other; if n is odd,
then be has one real nth root
2. b=0: 0 is the only nth root of b=0
3. b<0: If n is even, then b has no real nth root;
if n is odd, then b has one real nth root
Examples
• a^2=4 a= + or – 2
• a^2= -4 a= undefined
Rational Exponents and Radicals
• Notation
• b^1/2
• n square root of b
Definition 3: Principal nth Root
• For n a natural number and b a real number,
the principal nth root of b, denoted by b^1/n
or n square root of b is:
1. The real nth root of b if there is only one
2. The positive nth root of b if there are 2 real
nth roots
3. Undefined if b has no real nth root
b^m/n and b^-m/n
Rational number Exponent
• For m and n natural numbers and b only real
number
• b^m/n= (b^1/n)^m
• b^-m/n= 1/b^m/n
– Example: 4^(-3/2) = 1/(4^3/2)= 1/8
Simplifying Radicals
• Properties of Radicals
– Theorem 4
• For n a natural number greater than 1, and x and y
positive real numbers
– Properties
1. Nth root of square root of x^n= x
2. Nth root of square root of xy = nth root of x nth root of y
3. Nth root of squre root of x/y= nth square root of square
root of x over nth root of square root of y
Simplified form
1. No radical contains a factor to a power
greater than or equal to the index of the radical
2. No power of the radicand and the index of
the radical have a common factor than 1
3. No radical appears in a denominator
4. No fraction appears with in a radical
Rationalizing the denominator
• Eliminating a radical from the denominator
• Multiply the numerator and denominator by a
suitable factor that will leave the denominator
free of radicals
Rationalizing factor
• The suitable factor that leaves the
denominator free of radicals