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MAT 120 Chapter 10
Section 10.1: Radical Expressions & Functions
2
The number c is a square root of a if c  a .
Principle square root is always a ___________________________.
Simplify.
25
49
64
132
 4 
x8
x6
 36
 x  2
2
8
3
27
3
y3
3
 x6
2
x 2  8x  16
x10 y12 z 4
3
The number c is a cube root of a if c  a . The
3
0.0144
3
3
a c.
3
64
3
z9
8
3
27x15
MAT 120 Chapter 10
nth roots.
n
The number c is a nth root of a if c  a . The
5
32
5
32
7
y7
6
x
n
 4 81
6
Section 10.2: Fractions as exponents.
Review exponential Rules.
Zero Exponent Rule:
Product Rule:
Quotient Rule:
Power Rule:
Negative Exponent Rule:
9
 x  1
9
a c.
root
radicand
4
81
10
 x  4
10
MAT 120 Chapter 10
Rational Exponent Rule: Fractional Exponents
Convert the radicals to rational exponents.
3
x2
4
2xy 3
35 a 2b 4
Convert the expressions with rational exponents to radicals.
7
5x 
1
6
1
3
3xy
3
4
Simplify by using all exponential properties and positive exponents.
1
5
3 3
3
5
x
x
5
6
1
2

y


3
4




2
3


x y 




4

5
2
3
1
2
MAT 120 Chapter 10
Simplifying the radical expressions by using rational exponents.
6
2x 3
5
4
4 3
x
 a b
y 20
3
x
Section 10.3: Multiplying Radicals
*Multiplying Radical Expressions
*Simplifying Radicals by Factoring.
*Multiply and simplify….WRONG!
The Product Rule for Radicals.
n
a  n b  n ab
Multiply
3
2 3 7
5
3x 2  5 4 x
x2 x2
4
11 4 y

x
5
2
12
MAT 120 Chapter 10
Simplifying Radicals using the product rule and prime factorization.
675 x 2 y 5
72
3
11340a13b 20
135
3
 1500 x 3 z11
4
432r 13t 11
3x 2  6 x  3
5
 800 g 4 h8
6
256r 21t 24
4
24 x 3 y 2  4 10 x 5 y 3
Multiplying Radicals
15  6
23 25  33 20
MAT 120 Chapter 10
Section 10.4: Dividing Radicals
*Dividing Radical Expressions
*Simplifying Radicals by Factoring.
*Divide and simplify….WRONG!
The Quotient Rule for Radicals.
n
n
a n a

b
b
Divide
72 x 3 y
53 32
3
4
80
10
4
4
3 2x
Rationalizing the Denominator.
7
2
20
15
18a 9b 5
3
5
16
3a 2b
MAT 120 Chapter 10
Rationalizing the Denominator.
8x
3
12 xy 2
5
3y
4
24 x 3 y 2
5
7c
2 a 3b 2
Section 10.5: Several Radical Terms
* Adding and Subtracting Radicals
* Products or Quotients of 2+ Radicals
* Rationalizing Denominators with 2 terms
* Radicals with different roots(indices)
Adding and Subtracting Radicals. Combine Like Terms rules!
Combine Like Terms
Combine Like Radicals ________________
1.
1.
2.
2.
Simplify
3x  5 y  3  2 y  12 x  7
180  2 3  3 5  48
MAT 120 Chapter 10
Simplify
3
54  3 16  2 2
5 3x  7 x 2  3x 2  3 12 x
73 2 x 4 y 8  6 x3 16 y 5  2 y 2 3 2 x 7
Products ( Distributive Property & F.O.I.L )


3 2

3 x 5
4

3 25 2 6
3 8


 10  3  10  3 
MAT 120 Chapter 10
Rationalizing the Denominator
3 5 3
5 3
4
3x
Multiply & Divide with different roots.
5
x3  3 x
4
a 2b3  3 ab 2
3
x4
x
4
a 5b 3
6
7
a 3b 4
Section 10.6: Solving Radical Equations.
Principle of Powers
If a = b, then an = bn for any exponent n.
Use caution….false statements can become true statements!
x  22  4 x  23
x  y 5
4
x  y 3
MAT 120 Chapter 10
We use this principle to remove radicals.
3
x  5
4
x 2
Steps to solve radical equations.
1. Isolate the radical to one side of the equation.
________ check for _______ roots!
x  7
2 x  3  17  9
2. Remove the radical by the Principle of Powers.
*Steps 1 & 2 may have to be repeated.
3. Solve for x.
4. Always check your answer.
Solve for x.
3 x  7 1
x  x7 5
2 x  53  6  3
1
MAT 120 Chapter 10
Solve for x.
2x  5  1  x  3
x5  2 x7
MAT 120 Chapter 10
Section 10.7: Formulas – Pythagorean Theorem, Special Right Triangles, Midpoint
and Distance Formulas.
Pythagorean Theorem. In any right triangle, if a and b are the lengths of the legs
and c is the length of the hypotenuse, then a2 + b2 = c2.
Principle of Square Roots.
Solve for the missing side.
10
90
Special Right Triangles. 45o – 45o – 90o
Solve for x and y.
y
6
x
8
y
x
MAT 120 Chapter 10
Special Right Triangles. 30o – 60o – 90o
Solve for x and y.
y
x
y
8
12
x
Complete the table. The missing sides are opposite the angle in the column header. Leave your answer in
radical form.
60o
45o
o
30
30o
60 o
45o
90 o
9 cm.
45o
45 o
90 o
7 cm.
15 cm.
10 cm.
12 cm.
6 3 cm.
12 cm.
8 2 cm.
MAT 120 Chapter 10
Distance Formula & Midpoint Formula
Find the distance and the midpoint between (-3, 5) and (9, -11).
If M(3, 7) is the midpoint of segment AB, find the location of B when A is located
at (-6, 10).
MAT 120 Chapter 10
Section 10.8: Complex Numbers.
Define and simplify i to exponents.
Simplify radicals with negatives.
3
4
 5   20
Definition of Complex Numbers.
  11
 16   49
  50
MAT 120 Chapter 10
Add and Subtract Complex Numbers
12  3i   2  7i 
Multiply Complex Numbers
2i5  3i 
6  5i    9  8i 
4  5i 3  2i 
3  2i 3  2i 
Complex Conjugate Product Rule.
Divide Complex Numbers _________________________________________
2  3i
5  2i
7  4i
5i