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Algebra 2 Notes
Name: ________________
Section 6.6 – Radical Expressions & Radical Exponents
You are probably familiar with finding the square and square root of a number. These two operations are
inverses of each other. Similarly, there are roots that correspond to larger powers.
5 and 5 are _________________ roots of 25 because 52  25 and  5  25 .
2
2 is the _________________ root of 8 because 2 3  8 .
4
2 and 2 are _________________ roots of 16 because 2 4  16 and  2   16 .
a is the _________________ roots of b if a n  b .
The n th root of a real number a can be written as the radical expression _________________, where
n is the _________________ of the radical and a is the _________________. When a number has
more than one real root, the radical sign indicates only the _________________, or positive, root.
Case
Odd index
Even index; positive radicand
Even index; negative radicand
Radicand of 0
Numbers and Types of Real Roots
Roots
Example
1 real root
The real 3rd root of 8 is 2.
2 real roots
The real 4th roots of 16 are
0 real roots
-16 has no real 4th root.
1 root of 0
The 3rd root of 0 is 0.
Example 1:
Find all real roots.
a. fourth roots of 81
b. cube roots of -125
The properties of square roots also apply to
Properties of
For a  0 and
a.
3
27x6
n th roots.
n th Roots
b  0,
Words
Product Property of Roots
The n th root of a product is equal to
the product of the n n roots.
Quotient Property of Roots
The n th root of a quotient is equal to
the quotient of the n th root.
Example 2:
c. square root of –25
Numbers
3
Algebra
16  3 8  3 2  2 3 2
n
ab  n a  n b
25
25 5


16
16 4
n
a na

b nb
Simplify each expression. Assume that all variables are positive.
b.
4
3
x8
c.
3
x3
7
d.
3
x7  3 x2
 2.
A ____________________ exponent is an exponent that can be expressed as _______________, where
m and n are integers and n  0 . ____________________ expressions can be written by using rational
exponents.
Rational Exponents
For any natural number n and integer
Words
m,
Numbers
1
4
1
The exponent
indicates the n th root.
n
m
The exponent
indicated the n th root
n
raised to the m th power.
Example 3:
a.
 125
a.
7
1
n
16  16  2
2
83 
a na
4
 
3
2
8
m
 22  4
an 
 a
n
m
 n am
Write the expression in radical form and then simplify.
2
3
Example 4:
4
Algebra
1
b.
1
5
c.
64 3
d.
42
32 5
Rewrite each expression using rational exponents. Then simplify when possible.
b.
3
3
116
c.
4
252
Rational exponents have the same properties as integer exponents. Look back in Section 1.5 in your book if
you need to remember…
Example 5:
a.
3
5
25  25
Simplify each expression.
2
5
3
1
b.
83
2
83
c.
1
36 8  36 8