Download Unit 9 – Roots, Radicals, and Root Functions

Unit 9 – Roots, Radicals, and Root Functions
Unit 9.1 – Radical Expressions and Graphs
a  b means bn  a .
In the root, a is the radicand, n is the index, and the symbol is known as the radical sign.
n
Remember that numbers have 2 different roots. For example, 2 and -2 are both square roots of 4.
They are also fourth roots of 16. We now introduce the concept of the principal root.
nth Root
If n is even and a is positive, then
n
a is the principal nth root of a, and
 n a is the negative nth root of a.
If n is even and a is negative, then
n
a is not a real number.
If n is odd, then
there is exactly one nth root of a, written n a .
(If n is even, then the two roots of a are often written  n a .)
Graphing Functions Defined with Radicals
f x   x
10
8
6
4
2
-10 -8 -6 -4 -2
2
4
6
8 10
-2
-4
-6
-8
-10
f x   3 x
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Unit 9 – Roots, Radicals, and Root Functions
10
8
6
4
2
-10 -8 -6 -4 -2
2
4
6
8 10
-2
-4
-6
-8
-10
Simplifying Square Roots Using Absolute Values
n
an
If n is an even positive integer,
n
an  a ,
And if n is an odd positive integer,
Examples:
n
an  a .
72  7  7
 7 
2
 7  7
6
64  2
5
243  3
Use a calculator to find roots.
Demonstrate how to use calculator to find roots other than square roots. Either store value in x
and then use MATH menu or enter the index and then use MATH menu.
Unit 9.2 – Rational Exponents
Use the laws of exponents to resolve what this means:
2
1
1
 12 
3   32  32
 
 
3
1 1

2 2
but
 3
2
1
 3 therefore 3 2  3
 31
3
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Unit 9 – Roots, Radicals, and Root Functions
General rule:
1
If n a is a real number, then a n  n a .
Examples:
1
1
64 3  3 64  4
100 2  100  10
1
 256 4  4 256  4
Unit 9.3 – Simplifying Radical Expressions
Product Rule for Radicals
If n a and n b are real numbers and n is a natural number, then n a  n b  n ab . (Remember that
the product rule can only be used when the indexes are the same.)
Examples:
5  7  35
3
3  3 12  3 36
6
10m 4  6 5m  6 50m5
4
2  5 2 cannot be simplified, different indexes
Quotient Rule for Radicals
If
n
a and
n
b are real numbers and n is a natural number, then
n
a na
.

b nb
Examples:
16
16 4


25
25 5
3
8
8
8 2
2
3
 3


125
125
5
125 5
3

3
3
m6
m6 m 2


125 3 125
5
Simplifying Radicals
When is a Radical considered simplified?
1. The radicand has no factor raised to a power greater than or equal to the index.
2. The radicand has no fractions.
3. No denominator has a radical.
4. Exponents in the radicand and the index of the radical have no common factors (except 1).
Example:
To simplify 24 , first check to see if the radicand is divisible by a perfect square (the square of
a natural number) such as 4, 9, …. Choose the largest perfect square that divides in to 24, which
is 4. Write 24 as the product of 4 and 6, then use the product rule.
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Unit 9 – Roots, Radicals, and Root Functions
24  4  6  4  6  2 6
108 it may not be obvious that 108 is divisible by the perfect square 36. In such a case, factor
the radicand into its prime factors to aid in identifying perfect squares.
108  22  33  22  32  3  22  32  3  2  3  3  6 3
3
16 look for the largest perfect cube root that divides into 16.
3
16  3 8  2  3 8  3 2  2 3 2
Examples:
4
32
45
300
35
 3 54
243
Examples using radicals with variables:
16 x 3  16 x 2  x
 4x x
200 x 7 q8  200  x 7  q8 separate the individual elements
x 
 100  2 
3
2
x
q 
4
2
factor (square root; look for ways to factor squares)
=10  2  x3  x  q 4 take the perfect squares "outside" the radical sign
 10 x3 q 4 2 x commutative property
3
8 x 4 y 5 look for perfect cube(s), in this case 8 x3 y 3
Examples:
25 x 7
3
x5 y 7 z 6
 4 32a 5b7
Simplify Radicals by Using Smaller Indexes
Sometimes we can write a radical using rational exponents and then simplify the rational
exponent to lowest terms. Then we write the answer as a radical.
6
9
2
3
5  5  5  3 52  3 25
Examples:
9
6
12
23
6
t2
Multiplying Radicals with Different Indexes
Basically we rewrite the radical using rational exponents. Then we convert the rational
exponents into new exponents with a common denominator. Rewrite as radicals and use the
product rule.
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Unit 9 – Roots, Radicals, and Root Functions
7  3 2 common index is 6 (just like least common denominator)
1
3
1
2
7  7 2  7 6  6 73  6 343
3
2  2 3  2 6  6 22  6 4
7  3 2  6 343  6 4  6 1372 use the product rule for radicals
Example:
53 4
Unit 9.4 – Adding and Subtracting Radical Expressions
Simplify radical expressions involving addition and subtraction
4 2  3 2   4  3 2 use the distributive property
7 2
This is exactly the same process as 4 x  3x   4  3 x  7 x .
Only radical expressions with the same index and the same radicand can be combined.
Expressions such as 5 3  2 2 or 3 3  2 3 3 cannot be simplified by combining terms.
Example:
3 24  54  3 4  6  9  6 simplify each radical
 3  4  6  9  6 use the product rule for radicals
 3 2  6  3 6
 6 6 3 6
 9 6 distributive property
Examples:
3 5 7 5
2 11  11  3 44
5 12 x  6 75 x
9 5  4 10
Example:
2 3 16  5 3 54  2 3 8  2  5 3 27  2 factor perfect cubes
 2 3 8  3 2  5 3 27  3 2 product rule
 2  2  3 2  53 3 2
 4 3 2  15 3 2
 113 2 combine like terms
Examples:
7 3 81  3 3 24
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 2 4 32  7 4 162
3
p 4 q 7  3 64 pq
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Unit 9 – Roots, Radicals, and Root Functions
Adding and Subtracting Radicals with Fractions
Example:
75
8
25  3
42
2
4
2
4
16
32
16
16  2
 3    4   4  2 
16
 16  2 
 2  5  3   4  2   2 


4
 4  2 

 2 
25
5 3
 2 remove common factors from numerators and denominators
2
5 3 4

 write with common denominators
2
2
5 34

2

Examples:
5
4
10 3 6  3 3 9
x
x
80
81
 10
4
x
x
Unit 9.5 – Multiplying and Dividing Radical Expressions
Multiply binomial expressions by using the FOIL method.

5 3


6  1  5  6  5 1  3  6  3 1
 30  5  3 6  3
(this result cannot be simplified further)
Examples:
 7  3 
5 2


10  3

10  3


x y

x y

Rationalizing the denominator (with square roots)
This is the process of removing radicals from the denominator of a fraction.
Example:
3
3
7 3 7



7
7
7 7
It can be easier if you simplify the radicals before rationalizing.
6
6
6
3
3 3 3





 3
3
12
4 3 2 3
3 3
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Unit 9 – Roots, Radicals, and Root Functions
Examples:
3
48
200k 6
y7
72
y
Rationalizing denominators (with cube roots)
27
3
First use the quotient rule to simplify the numerator and denominator.
16
27 3 27
3
3
3
 3
3 3  3
16
16
8 2 2 2
To get a rational denominator, multiply the numerator and denominator by a number that will
result in a perfect cube in the radicand of the denominator. In this case, since 2  4  8 , multiply
the numerator and denominator by 3 4 .
3
3 34
33 4
33 4 33 4 33 4






4
23 2 23 2 3 4 23 2  3 4 23 8 22
Examples:
3
15
32
3
m12
n
Rationalize denominators with binomials involving radicals
Ask them to recall what happens when you multiply x  y x  y  : you get the difference of


squares x 2  y 2 .
We use the same process to simplify a denominator with radicals, such as
4
.
3 5
We call binomials such as 3  5 and 3  5 conjugates.
Unit 9.6 – Solving Equations with Radicals
An equation that includes one or more radical expressions with a variable is a radical equation.
3
x4 8
5x  4  3 2 x 1
6  x  27
Power Rule for Solving Equations with Radicals
If both sides of an equation are raised to the same power, all solutions of the original equation
are also solutions of the new equation.
BE CAREFUL: the power rule does not say that all solutions of the new equation are solutions
of the original. You must check every solution in the original equation to be sure.
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Unit 9 – Roots, Radicals, and Root Functions
Examples:
3x  4  8


3x  4
2
3  20  4  8
 82
3x  4  64 check this answer:
3x  60
60  4  8
64  8
88
x  20
Steps to solve an equation with radicals
1. Isolate the radical. Make sure that one radical term is alone on one side of the equation
2. Apply the power rule. Raise both sides of the equation to a power that is the same as the
index of the radical.
3. Solve. Solve the resulting equation; if it still contains a radical, repeat Steps 1 & 2.
4. Check all potential solutions in the original equation.
Examples:
5x  1  3  0
5 x  1  3

5x 1

2
5  2 1  3  0
  3 check your answer
2
5x 1  9
10  1  3  0
9 3 0
33  0
5 x  10
x2
3+3 obviously is not 0, so 2 is not a solution and the solution set is null.
4 x  x2

4 x

2
  x  2
2
4  x  x 2  4 x  4 note this is quadratic so get 0 on one side
0  x2  5x
0  x  x  5
x  0 or x  5  0
x  0 or x  5
Remember to check your answers in the original equation. In this case -5 does not work in the
original equation so the solution set is {0}.
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Unit 9 – Roots, Radicals, and Root Functions
5 x  6  3 x  4  2 isolate one radical, then square both sides
5 x  6  2  3x  4

5x  6
  2 
2
3x  4

2
5 x  6  4  4 3 x  4   3 x  4  there's still a radical so we'll have to square again
5 x  6  8  3 x  4 3 x  4 first we need to isolate the remaining radical
2 x  2  4 3 x  4
x  1  2 3 x  4 divide by common factor of 2
 x  1
2
  2 
2

3x  4

2
square both sides
x2  2 x  1  4  3x  4 
x 2  2 x  1  12 x  16 it's quadratic so set equal to 0
x 2  14 x  15  0
 x  15 x  1  0
x  15  0 or x  1  0
x  15 or x  1
Check both answers in the original equation to find that only -1 works. So the solution set is {1}.
How about powers greater than 2?
3
x  5  3 2x  6

3
x5
 
3
3
2x  6

3
raise both sides to the third power
x  5  2x  6
11  x
Checking this answer in the original equation shows the solution set is {11}.
Examples:
3x  5  x  1
3
x 2  3 x  12  3 x 2
4x2  2x  3  2 x  7
4
x 1  x  4  1
2x  5 1  0
Unit 9.7 – Complex Numbers
We are going to talk about a new set of numbers. A set that includes the real numbers, as well as
numbers that are even roots of negative numbers, such as 2 .
In order to do this, a new number was discovered or invented around the time of the Reformation
(1500’s). It is called the ‘imaginary unit’ and is denoted by i. It is thought it was called
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Unit 9 – Roots, Radicals, and Root Functions
“imaginary” because everyone ‘knew’ there was no use for such a number. It turns out that there
are uses in electrical engineering and other fields.
The imaginary unit is defined as the number such that i 2  1 so i  1 .
Using this we can define the square root of any negative number as
b for any positive b
b  i b
Examples:
100  i 100  10i
Examples:
16
 81
7
32
Multiplying square roots of negative numbers
The product rule for radicals does not work because it only applies to positive radicands.
4  9  i 4  i 9
 i  2i 3
 6i
Notice that the product rule does not work!
4  9  4  9
 36
6
4  9  4  9
Examples:
3   7  i 3  i 7
i
2
2  8  i 2  i 8
3 7
i
  1 21
2
  1 16
 4
  21
Dividing square roots of imaginary numbers
This follows a similar process.
Examples:
75 i 75
32 i 32


8
3
i 3
8

75
3
i
 i 56
 i 30
32
8
 25
i 4
5
 2i
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2 8
5  6  i 5  6
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Unit 9 – Roots, Radicals, and Root Functions
Examples:
7  7
32
2
5  10
15  2
40
10
Complex Numbers
Complex numbers are of the form a + bi, where a and b are real numbers. The number a is
called the real part and b is called the imaginary part. If a is 0 they are called pure imaginary
numbers.
Add and subtract complex numbers
We use the commutative, associative, and distributive properties to add and subtract complex
numbers.
To add, we add the real parts and add the imaginary parts.
Examples:
5   9  3i    5  9   3i
 2  3i    6  4i    2  6    3i  4i 
 8  7i
 14  3i
To subtract, we subtract the real parts and subtract the imaginary parts.
Examples:
 6  5i    3  2i    6  3   5i  2i 
 7  3i   8  6i    7  8   3i   6i  
 3  3i
 9  4i    9  8i    9   9     4i  8i 
 1  3i
 0  4i
 4i
Examples:
 7  3i    4  2i    7  4    3i  2i 
 6  i    5  4i    6  5   i  4i 
 3i
 1  3i
Multiplying complex numbers
To multiply complex numbers, treat them like polynomials and FOIL them.
Examples:
4i  2  3i   4i  2   4i  3i 
 3  5i  4  2i   3  4   3  2i   5i  4   5i  2i 
 8i  12i 2
 12  6i  20i  10i 2
 8i  12  1
 12  14i  10  1
 8i  12
 12  14i  10
 12  8i
 22  14i
Examples:
6i  4  3i 
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 6  4i  2  4i 
3  2i 3  2i 
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Unit 9 – Roots, Radicals, and Root Functions
Dividing complex numbers
The quotient of two complex numbers should be a complex number. We do need to eliminate
the i from the denominator. To do this we use the conjugate of the denominator.
Examples:
2i 2i 3i


3i 3i 3i
2  3  2  i   3  i   i 2
 2
3  3i   3i   i2

6  5i  1
9   1
5  5i
10
5 5i
 
10 10
1 1
  i
2 2
Examples:
8  4i
5
1 i
3  2i

5i
i
Finding powers of i
i1  i
The powers of i rotate through 4 numbers:
i 2  1
i 3  i 2  i  i
i 4  i 2  i 2  1   1  1
So, for example i 75   i 4   i 3  1  i 3  1 i 3  i 3  i which suggests a method to simplify
18
18
large powers of i:
Divide the exponent by 4, throw away the answer and just keep the remainder as the exponent of
i.
i 233
233  4  58 R1
 
i 233  i 4
58
 i1
 1 i
i
Examples:
i 21
i36
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i 50
i 9
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