Chapter 7: Radicals and Complex Numbers
Lecture notes
Math 1010
Section 7.1: Radicals and Rational Exponents
Definition of nth root of a number
Let a and b be real numbers and let n be an integer n ≥ 2. If a = bn , then b is an nth root of a. If n = 2, the
root is called square root. If n = 3, the root is called cube root.
Definition of principal nth root of a number
Let a be a real number that has at least one (real number) nth root. The
√ principal nth root of a is the nth root
that has the same sign as a and it is denoted by the radical symbol n a. The positive integer n is the index
of the radical, and a is the radicand.
Ex.1
Examples of
√nth roots.
9
(1) 3 = √
(2) −5 =√ 25
4
(3) 2 = √
16
(4) 4 = 3 64
Ex.2
Find each
√ principal root.
(1) √
36
(2) −
√ 36
(3) √−4
3
8
(4) √
(5) 3 −8
1
Chapter 7: Radicals and Complex Numbers
Lecture notes
Math 1010
Properties of nth roots
Let a be a real number.
√
(1) If √
a is positive and n is even, then a has exactly two real nth roots, which are denoted by n a and
− n a.
√
(2) If n is odd (a is any real number), then a has one real nth root, which is denoted by n a.
(3) If a is negative and n is even, then a has no (real) nth root.
Ex.3
√
√
(1) 81
has two real square roots: 9 = 3 and − 9 = −3.
√
3
(2) √
27 = 3
(3) −25 has no real square root.
Perfect squares and perfect cubes
A perfect square is an integer which is a square of an integer. A perfect cube is an integer which is a cube of an
integer.
Ex.4
State whether each number is a perfect square, a perfect cube, both, or neither.
(1) 81
(2) −125
(3) 64
(4) 32
(5) 1
2
Chapter 7: Radicals and Complex Numbers
Lecture notes
Math 1010
Properties of nth powers and nth roots
Let a be a real number and
√ n be an integer n ≥ 2.
(1) If n is odd, then ( n√a)n = a.
(2) If n is even, then ( n a)n = |a|.
Ex.5
Evaluate√each radical expression
(1) (p 5)2
(2) √ (−5)2
3
(3) p43
(4) p(−3)2
(5) −(32 )
Definition of rational exponents
Let a be a real number and let n be an integer such that n ≥ 2. If the principal nth root of a exists, then
√
1
an = n a
If m is a positive integer that has no common factor with n, then
√
√
m
m
1
1
a n = (a n )m = ( n a)m and a n = (am ) n = n am
Rules of Exponents
Let m and n be rational numbers, and let a and b represent real numbers, variables, or algebraic expressions,
a 6= 0, b 6= 0.
(1) am · an = am+n
m
(2) aan = am−n
(3) (ab)m = am · bm
m n
mn
(4) (a
) = a
(5)
a
b
0
m
=
am
bm
(6) a = 1
(7) a−m = a1m
−m m
(8) ab
= ab
3
Chapter 7: Radicals and Complex Numbers
Lecture notes
Ex.6
Evaluate each expression.
4
(1) 8 3
3
(2) 25− 2
64 23
(3) ( 125
)
1
(4) −16 2
1
(5) (−16) 2
4
Math 1010
Chapter 7: Radicals and Complex Numbers
Lecture notes
Ex.7
Rewrite √
each expression using rational exponents.
4
(1) x√ x3
3 2
(2) √xx3
p
(3) 3 x2 y
Ex.8
Use theprule of exponents to simplify each expression.
√
3
(1)
x
4
(2)
(2x−1) 3
√
3
2x−1
5
Math 1010
Chapter 7: Radicals and Complex Numbers
Lecture notes
Definition of radical function
A radical function is a function that contains a radical.
Ex.9
Evaluate each radical
function when x = 4.
√
(1) f (x) = √3 x − 31
(2) g(x) = 16 − 3x
Domain of a radical function
Let n be an integer, n ≥ 2.
√
• If n is odd, the domain of f (x) = n√x is the set of all real numbers.
• If n is even, the domain of f (x) = n x is the set of all non-negative real numbers.
Ex.10
Describe the domain
of each radical function.
√
(1) f (x) = √3 x
(2) g(x) = x3
6
Math 1010
Chapter 7: Radicals and Complex Numbers
Lecture notes
Ex.11
Find the domain of
f (x) =
√
Math 1010
2x − 1
Section 7.2: Simplifying Radical Expressions
Product and Quotient Rules for Radicals
Let u and v be real numbers, variables, or algebraic expressions. If the nth roots of u and v are real, the
following rules are true.
•
√
√ √
n
uv = n u n v
•
r
√
n
u
u
n
= √
, v 6= 0
n
v
v
Ex.1
Simplify
√ each radical by removing as many factors as possible.
(1) √12
(2) √48
(3) √75
(4) 162
7
Chapter 7: Radicals and Complex Numbers
Lecture notes
Ex.2
Simplify
√ each radical expression.
(1) √25x2
(2) √12x3
(3) √144x4
3
(4) √
40
5
(5) p486x7
3
(6) q
128x3 y 5
(7)
(8)
81
25
√
56x2
√
8
8
Math 1010
Chapter 7: Radicals and Complex Numbers
Lecture notes
Ex.3
Simplify
r
−
3
y5
27x3
Simplifying Radical Expressions
A radical expression is in the simplest form if
(1) All possible nth powered factors have been removed from each radical.
(2) No radical contains a fraction.
(3) No denominator of a fraction contains a radical.
Ex.4
Rationalize
q the denominator in each radical expression.
(1)
(2)
(3)
3
5
4
√
3
9
√8
3 18
9
Math 1010
Chapter 7: Radicals and Complex Numbers
Lecture notes
Math 1010
Ex.5
Rationalize
q the denominator in each radical expression.
8x
12y 5
(1)
(2)
q
3
54x6 y 3
5z 2
Ex.6
Find the length of the hypothenuse of the following right triangle
Ex.7
A softball diamond has the shape of a square with 60-foot sides. The catcher is 5 feet behind home plate.
How far does the catcher have to throw to reach second base?
10
Chapter 7: Radicals and Complex Numbers
Lecture notes
Math 1010
Section 7.3: Adding and Subtracting Radical Expressions
Like Radicals
Two or more radical expressions are like radicals if they have the same index and the same radicand.
Ex.1
Simplify
radical√expression by combining like radicals
√ each √
7+5 √
7 − 2 √7
(1) √
√
(2) 3√3 x + 2 3√x + x − 8 x
(3) 45x + 3√ 20x
√
(4) 5√3 x − x√ 4x
√
3
3
(5) 6x4 + 3 48x − 162x4
11
Chapter 7: Radicals and Complex Numbers
Ex.2
Simplify
Lecture notes
√
5
7− √
7
Section 7.4: Multiplying and Dividing Radical Expressions
Ex.1
Find each
and simplify
√
√ product
6· √
3
(1) √
3
3
(2) 3
16
√ 5· √
(3) √3(2 + √5)
(4) √2(4
√− 8)√
(5) 6( 12 − 3)
12
Math 1010
Chapter 7: Radicals and Complex Numbers
Lecture notes
Ex.2
Find the product
and
√
√ simplify
(1) (2 7√− 4)( 7√
+ 1)
(2) (3 − x)(1 + x)
Ex.3
Find each conjugate
of the expression and multiply the expression by its conjugate
√
(1) 2√− 5
√
(2) 3 + x
13
Math 1010
Chapter 7: Radicals and Complex Numbers
Lecture notes
Ex.4
Simplify
(1)
√
1−
(2)
3
√
5
4
√
2− 3
(3)
√
√
5 2
√
7+ 2
14
Math 1010
Chapter 7: Radicals and Complex Numbers
Lecture notes
Ex.5
Perform each
√ division and simplify
(1) 6 ÷ (√x − 2)√
√
(2) (2 − √3) ÷√
( 6 + 2)
(3) 1 ÷ ( x − x + 1)
15
Math 1010
Chapter 7: Radicals and Complex Numbers
Lecture notes
Math 1010
Section 7.5: Radical Equations and Applications
Raising each side of an equation to the nth power
Let u and v be numbers, variables, or algebraic expressions, and let n be a positive integer. If u = v, then it
follows that un = v n . This is called raising each side of an equation to the nth power.
Ex.1
Solve
Ex.2
Solve
√
√
x−8=0
3x + 6 = 0
16
Chapter 7: Radicals and Complex Numbers
Ex.3
Solve
Ex.4
Solve
Lecture notes
√
3
√
2x + 1 − 2 = 3
5x + 3 =
17
√
x + 11
Math 1010
Chapter 7: Radicals and Complex Numbers
Ex.5
Solve
Ex.6
Solve
Lecture notes
√
4
3x +
√
√
4
2x − 5 = 0
x+2=x
18
Math 1010
Chapter 7: Radicals and Complex Numbers
Ex.7
Solve
Lecture notes
√
3x + 1 = 2 −
√
Math 1010
3x
Section 7.6: Complex Numbers
The square root of a negative number
Let c be a positive real number. Then the square root of −c is given by
p
√
√ √
√
−c = c(−1) = c −1 = ( c)i
Ex.1
Write each
√ number in i-form.
(1) q−36
(2)
(3)
(4)
√
− 16
25
−54
√
√−48
−3
19
Chapter 7: Radicals and Complex Numbers
Lecture notes
Math 1010
Ex.2
Perform
√each operation.
√
(1) √−9 + −49
√
(2) −32 − 2 −2
Ex.3
Find each
√ product.
√
(1) √−15√ −15 √
(2) −5( −45 − −4)
Definition of complex number
A number of the form a + bi, where a and b are real numbers, is called a complex number, and it is said to
be written in standard form. The real number a is called the real part and the real number b is called the
imaginary part of the complex number a + bi. If b = 0, the number a + bi = a is real. If b 6= 0, the number
a + bi is called imaginary. If a = 0, the number a + bi is called pure imaginary number.
20
Chapter 7: Radicals and Complex Numbers
Lecture notes
Ex.4
√
√
√
Determine whether the complex numbers −9 + −48 and 3 − 4 3i are equal.
Ex.5
Find the values of x and y that satisfy the equation
√
3x − −25 = −6 + 3iy
21
Math 1010
Chapter 7: Radicals and Complex Numbers
Lecture notes
Ex.6
Perform each operation and write the result in standard form.
(1) (3 − i) + (−2 + 4i)
(2) 3i + (5 − 3i)
(3) 4 − (−1 + 5i) +√(7 + 2i)√
(4) (6 + 3i) + (2 − −8) − −4
Ex.7
Perform each operation and write the result in standard form.
(1) (7i)(−3i)
√
(2) (1 − i)( −9)
(3) (2 − i)(4 + 3i)
(4) (3 + 2i)(3 − 2i)
22
Math 1010
Chapter 7: Radicals and Complex Numbers
Lecture notes
Ex.8
Write each quotient of complex numbers in standard form.
(1) 2−i
4i
5
(2) 3−2i
(3) 8−i
8+i
2+3i
(4) 4−2i
23
Math 1010