Section3: Rational Exponent and Radicals
 Roots
Definition: nth Roots
If n is appositive integer and an = b, then a is called an nth
root of b. if a2 = b, then a is a square root of b. if a3 = b, then a
is the cube root of b.
Definition: Exponent 1/n
If n is a positive even integer and a is positive ,then a1/n
denotes the positive real nth root of a and is called principal
nth root of a.
If n is a positive odd integer and a is any real number ,then
a1/n denotes the real nth root of a.
If n is appositive integer then 0 1/n=0
Example(1)
Evaluate each expression.
a.41/2
b.81/3
C.(-8)1/3 D.(-4)1/2
Solution
T. EZDIHAR
Page 1
 Rational Exponent
Definition :Rational Exponent
If m and n are positive integer, then
am/n =(a1/n)m
provided that a1/n is a real number.
Example(2)
Evaluate each expression
a.(-8)2/3
b.27-2/3
c.1006/4
Solution
Rule for Rational Exponents
The following rules are valid for all real numbers a and b and
rational numbers r and s, provided that all indicated powers
are real and no denominator is zero.
T. EZDIHAR
Page 2
1. ar as = ar+s
2.ar/as=ar-s
3.(ar )s = ar s
4.(ab)r = arbr
s
=bs/as
5.(a/b)r = ar/br 6.(a/b) –r =(b/a)r 7. a-r/b-
Example (3):
Simplify each expression, using absolute value when
necessary Assume that the variable can represent any real
numbers.
a.(64 a6)1/6
b.(x9)1/3
c.(a8)1/4
d.(y12)1/4
Solution
Example(4):
Use the rules of exponents to simplify each expression
.Assume that the variables represent positive real numbers
.Write answers without negative exponents.
a. x 2/3 x4/3
b.(x4 y1/2)1/4
c.(a3/2 b2/3/a2)3
Solution
T. EZDIHAR
Page 3
 Radical Notation
Definition: Radical
If n is appositive integer and is a number for which a1/n is
𝑛
defined, then the expression √𝑎 is called a radical ,and
𝑛
√𝑎= a1/n
2
If n=2 we write √𝑎 rather than √𝑎 .
Example 5:
Evaluate each expression.
a.√49
3
b. √−1000
c. 4√16/81
Solution
T. EZDIHAR
Page 4
Rule :Converting a m/n to Radical Notation
𝑛
If a is a real number and m and n are integers for which √𝑎
is real, then
𝑛
𝑛
am/n= ( √𝑎 )m= √𝑎𝑚
Example(6):
Write each expression in radical notation .Assume that all
variables represent positive real numbers .Simplify the
radicand if possible.
a.2 2/3
b.(3x) 3/4
c.2(x2+3)-1/2
Solution
T. EZDIHAR
Page 5
 The Product and Quotient Rules for Radical
Rule: Product and Quotient
For any positive integer n and real numbers a and b (b≠0)
𝑛
𝑛
𝑛
1. √𝑎𝑏 = √𝑎 . √𝑏
𝑛
𝑛
2. 𝑛√𝑎/𝑏 = √𝑎 / √𝑏
Product rule for radicals
Quotient rule for radicals
Example(7):
Simplify each radical expression .Assume that all variables
represent positive real numbers.
3
a. √125𝑎6
T. EZDIHAR
b.√3/16
5
c.√−32𝑦 5 /𝑥 20
Page 6
 simplified form and rationalizing the denominator.
Definition: simplified Form for Radicals of Index n.
A radical of index n in simplified form has
1. No perfect nth powers as factors of the radicand.
2. No fractions in the radicand, and
3. No radicals in a denominator.
Example(8)
Write each radical expression in simplified form .Assume
that all variables represent positive real numbers.
a.√20
b. √24𝑥 8 𝑦 9
c.9/√3
3
d.√3/5𝑎4
Solution
T. EZDIHAR
Page 7
 Operation with Radical Expressions
2√7 + 3√7 = 5√7
Example(9)
Perform each operation and simplify the answer. Assume
that all variable represents positive real number.
a.√20 + √5
3
3
b. √24𝑥 - √81𝑥
4
4
c. √4 𝑦 3 . √12 𝑦 2
d.√40 ÷ √5
Solution
T. EZDIHAR
Page 8
Example:
write each expression using a single radical symbol.
Assume that each variable represents appositive real
number.
3
a) √2 . √3
b) 3√𝑦 . 4√2𝑦
3
c) √ √2
T. EZDIHAR
Page 9
Theorem: mth root of an nth root.
If m and n are positive integers for which all of the
𝑚
𝑛
𝑚𝑛
following roots are real, then √ √𝑎 = √𝑎.
T. EZDIHAR
Page 10