Math 1014: Precalculus with Transcendentals
Ch. P: Prerequisites: Fundamental Concepts of Algebra
Sec. P3: Radicals and Rational Exponents
I.
Radicals
A. Square Roots
1. Definition of the Principal Square Root:
If
a is a nonnegative real number, the nonnegative number b such that b 2 = a ,
denoted by
b = a , is the principal square root of a.
radical sign →
a ←radicand
a. Examples
1)
9=
2)
− 9=
3)
−9 =
b. The symbol
−
is used to denote the negative square root of a number.
c. Note: In the real number system, negative numbers do NOT have square roots.
2. Square Roots of Perfect Squares
a. For any real number a,
a2 = a .
b. Examples
1)
32 =
2)
(−3)2 =
3. Product Rule for Square Roots
a. If a and b represent nonnegative real numbers, then
ab = a ⋅ b and
b. Examples
1)
400 =
2)
700 =
3)
3 ⋅ 12 =
4)
3x ⋅ 15x =
a ⋅ b = ab .
4. Quotient Rule for Square Roots
a. If a and b represent nonnegative real numbers and
a
a
=
and
b
b
b ≠ 0 , then
a
a
=
.
b
b
b. Examples
1)
49
=
25 __________________________________________________________________________________________
2)
200x 3
=
10x −1 __________________________________________________________________________________________
5. Adding and Subtracting Square Roots: simplify like radicals
Examples
1)
14 3 − 9 3 =
2)
8 32x + 2 50x =
3)
3 54 − 2 24 − 96 + 4 63 =
_____________________________________________________________________________________
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6. Rationalizing the Denominator
a. Denominator Consists of a Single Square Root Term
If the denominator consists of the square root of a natural number that is not a
perfect square, multiply the numerator and the denominator by the smallest
number that produces the square root of a perfect square in the denominator.
Examples: Rationalize the denominator and simplify:
1)
5
=
7 __________________________________________________________________________________________
2)
8
=
12 __________________________________________________________________________________________
b. Denominator Consists of the Sum or Difference of Radical Expressions
If the denominator consists of radical expressions that involve the sum or
difference of two terms, multiply the numerator and the denominator by the
conjugate of the denominator.
Examples: Rationalize the denominator and simplify:
1)
3
=
5 + 2 __________________________________________________________________________________________
2)
7
=
2 3 − 7 __________________________________________________________________________________________
B. Other Kinds of Roots
1. Definition of the Principal
*
n
nth Root of a Real Number
a = b means that b n = a .
If n, the index, is even, then
nonnegative
If
a must be nonnegative ( a ≥ 0 ) and b is also
(b ≥ 0 ) .
n is odd, then a and b can be any real numbers.
Examples
a.
4
16 =
b.
3
−64 =
2. Finding nth Roots of Perfect
a. Rule
If
n is odd,
If
n is even,
an = a .
n
n
b. Examples
1)
3
(−5)3 =
2)
4
(−5)4 =
3)
6
−49 =
an = a .
nth Powers
2. Product and Quotient Rules for
nth Roots
a. Product Rule
For all real numbers
n
a and b, where the indicated roots represent real numbers,
ab = n a ⋅ n b and
n
a ⋅ n b = n ab
b. Quotient Rule
For all real numbers
n
a na
=
and
b nb
a and b, where the indicated roots represent real numbers,
n
n
a na
=
b
b
c. Examples
1)
3
4 ⋅ 3 20 =
4
512x 5
2)
4
3)
II.
3
2x −3
=
__________________________________________________________________________________________
24xy 3 − y 3 81x =
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Rational Exponents
A. Definition of
1. If n
a
1
an
represents a real number, where
2. Furthermore,
B. Definition of
1. If
n
a− n =
1
1
1
1
, a≠0
1 =
n
a
an
m
an
a represents a real number and
a n = n am = ( n a )
m
2. Furthermore, if
a− n =
m
n ≥ 2 is an integer, then a n = n a .
a− n
m
1
m , a ≠ 0
an
m
m
is a positive rational number, n ≥ 2 , then
n
.
is a nonzero real number, then
C. Examples: Simplify
1.
32
2.
6
3.
− 25
x4 =
100x 4
5
5x 2
4.
=
=
____________________________________________________________________________________________________
1
9 6 4 3
(125x y z )
=