Post on 03-Jan-2016
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Section 9.1
Finding Roots
OBJECTIVES
Find the square root of a number.
A
Square a radical expression.
B
OBJECTIVES
Classify the square root of a number and approximate it with a calculator.
C
OBJECTIVES
Find higher roots of numbers.
D
Solve an application involving square roots.
E
DEFINITION
If a is a positive real number,
a = b is the positive square root of a so that b2 = a.
– a = b is the negative square root of a so that b2 = a.
Square Root
DEFINITION
If a = 0, a = 0 and 0 = 0since 02 = 0
Square Root
When the square root of a non-negative real number a is squared, the result is that positive real number:
RULESquaring a Square Root
2 2= and – = .a a a a
RULE
If a is negative, a isnot a real number.
Square Root of aNegative Number
Section 9.1Exercise #1
Chapter 9Roots and Radicals
Find.
a) 169 b) – 49
81
= 132
= 13
= – 72
92
= – 7
9
Section 9.1Exercise #2
Chapter 9Roots and Radicals
Find the square of each radical expression.
a) – 121 b) x2 + 7
2 = – 121
= 121
22 = + 7
x
= x 2 + 7
Section 9.1Exercise #3
Chapter 9Roots and Radicals
Classify each number as rational, irrational, or not a real number, and simplify if possible.
a) 17 b) – 36
irrational
Not real
= – 6
=
10
7
c) –100 d) 100
49
rational
rational
Section 9.1Exercise #4
Chapter 9Roots and Radicals
Find each root, if possible.
a) 814 b) – 6254
= 344 = – 544
= 3 = – 5
Section 9.1Exercise #5
Chapter 9Roots and Radicals
t =
d
5
d = 20 t =
205
t = 4 = 2
The diver takes 2 seconds.
A diver jumps from a cliff 20 meters high. If the time t (in seconds) it takes an object dropped from a distance
d (in meters) to reach the ground is given by: How long does it take the diver to reach the water?
Section 9.2
Multiplication and Division of Radicals
OBJECTIVES
Multiply and simplify radicals using the product rule.
A
OBJECTIVES
Divide and simplify radicals using the quotient rule.
B
OBJECTIVES
Simplify radicals involving variables.
C
OBJECTIVES
Simplify higher roots.D
If a and b are nonnegative numbers,
Product Rule for Radicals
= ba b a
If a and b are positive numbers,
Quotient Rule for Radicals
ab
= ab
For any real number a,
Absolute Value for Radicals
a2 = a
For all real numbers where the indicated roots exist,
Properties of Radicals
= and = nn nn nn
a aa b a bb b
Section 9.2Exercise #6
Chapter 9Roots and Radicals
Simplify.
a) 125 b) 54
= 25 • 5
= 5 5
= 9 • 6
= 3 6
Section 9.2Exercise #7
Chapter 9Roots and Radicals
Multiply.
a) 3 • 11 b) 11 • y , y > 0
= 3 • 11
= 33
= 11 • y
= 11y
Section 9.2Exercise #8
Chapter 9Roots and Radicals
Simplify.
a)
7
16 b)
21 50
7 5
=
7
16
=
7
4
=
21
7 •
50
5
= 3 10
Section 9.2Exercise #9
Chapter 9Roots and Radicals
Simplify.
a) 144n2 , n > 0 b) 32y7 , y > 0
= 122n2
= 12n
= 42 • 2 • y6 • y
= 4y3 2y
Section 9.2Exercise #10
Chapter 9Roots and Radicals
Simplify.
a) 964
= 16 • 64
= 24 • 64
= 2 64
Section 9.3
Addition and Subtractions of Radicals
OBJECTIVES
Add and subtract like radicals.
A
OBJECTIVES
Use the distributive property to simplify radicals.
B
OBJECTIVES
Rationalize the denominator in an expression.
C
Rationalizing Denominators
PROCEDURE
Method 1:
Multiply both numerator and denominator of the fraction by the square root in the denominator.
Rationalizing Denominators
PROCEDURE
Method 2: Multiply numerator and denominator by the square root of a number that makes the denominator the square root of a perfect square.
Section 9.3Exercise #11
Chapter 9Roots and Radicals
Simplify.
a) 9 13 + 7 13 b) 14 6 – 3 6
= 9 + 7 13
= 16 13
= 1 4 – 3 6
= 11 6
Section 9.3Exercise #12
Chapter 9Roots and Radicals
Simplify.
a) 28 + 63
= 4 7 + 9 7
= 2 7 + 3 7
= 2 + 3 7
= 5 7
Section 9.3Exercise #13
Chapter 9Roots and Radicals
Simplify.
3 a) 18 – 5 • = 3 18 – 3 • 5
= 3 • 9 • 2 – 3 • 5
= 9 • 3 • 2 – 3 5
= 3 6 – 15
Section 9.3Exercise #14
Chapter 9Roots and Radicals
=
3 • 5
20 • 5
Write
3
20 with a rationalized denominator.
=
15
100
=
15
10
Section 9.3Exercise #15
Chapter 9Roots and Radicals
=
y 2 • 2
50 • 2
Write
y 2
50, y > 0 with a rationalized denominator.
=
y 2
100
=
y 2
10
Section 9.4
Simplifying Radicals
OBJECTIVES
Simplify a radical expression involving products, quotients, sums, or differences.
A
OBJECTIVES
Use the conjugate of a number to rationalize the denominator of an expression.
B
OBJECTIVES
Reduce a fraction involving a radical by factoring.
C
Simplifying Radical ExpressionsRULES
1. Whenever possible, write the rational-number representation of a radical expression.
34 2 1 181 as 9, as , and as 9 3 8 2
Simplifying Radical ExpressionsRULES
2. Use the product rule x • y = xy to write indicated products as a single radical.
Simplifying Radical ExpressionsRULES
6 instead of 2 • 3 and 2ab instead of 2a • b
Section 9.4Exercise #16
Chapter 9Roots and Radicals
= 8 14 – 14 a) 8 14 – 7 • 2
Simplify.
= 8 – 1 14
= 7 14
Simplify.
b) 12x 3
4x 2, x > 0
=
12x 3
4x 2
=
12
4 • x3 – 2
= 3x
Section 9.4Exercise #17
Chapter 9Roots and Radicals
=
500
23
a)
5003
23
Simplify.
= 2503
= 125 • 23
= 5 23
Section 9.4Exercise #18
Chapter 9Roots and Radicals
10 – 2 20 10 + b) 2 20
Simplify.
2 2 = 10 – 2 20
= 10 – 4 20
= 10 – 80
= – 70
Section 9.4Exercise #19
Chapter 9Roots and Radicals
a)
11
3 + 1
Simplify.
3 – 111 •
=3 + 1 1 3 –
11 3 – 1 • =
3 – 1
=
11 3 – 11
2
Section 9.4Exercise #20
Chapter 9Roots and Radicals
a)
– 6 + 18
3
Simplify.
=
– 6 + 9 • 2
3
=
– 6 + 3 2
3
3 – 2 + 2 =
3
= – 2 + 2
Section 9.5
Applications
OBJECTIVES
Solve equations with one square root term containing the variable.
A
OBJECTIVES
Solve equations with two square root terms containing the variable.
B
OBJECTIVES
Solve an application.C
Raising Both Sides of an Equation to a Power
If both sides of the equation A = B
are squared, all solutions are
among the solutions of the new
equation A2 =B2.
PROCEDURE
Solving Radical EquationsPROCEDURE
1. Isolate the square root term containing the variable.
2. Square both sides of the equation.
Solving Radical EquationsPROCEDURE
3. Simplify and repeat steps 1 and 2 if there is a square root term containing the variable.
Solving Radical EquationsPROCEDURE
4. Solve the resulting linear or quadratic equation.
5. Check all proposed solutions in the original equation.
Section 9.5Exercise #22
Chapter 9Roots and Radicals
x + 4 – x = 2
Solve.
x + 4 = x + 2
x + 4 = x 2 + 4x + 4
0 = x 2 + 3x
0 = + 3x x
x = 0 or x = – 3
Solve.
x = 0 or x = – 3
2 – 0 = 2
4 2 1 + 3 ? 2
Check: x = – 3, – 3 + 4 – – 3 ? 2
Check: x = 0, 0 + 4 – 0 ? 2
Section 9.5Exercise #23
Chapter 9Roots and Radicals
y + 3 = 2y + 1
Solve.
y + 3 = 2y + 1
3 = y + 1
Check: 2 + 3 ? 2 • 2 + 1
5 ? 4 + 1
5 = 5
2 = y y = 2
Section 9.5Exercise #24
Chapter 9Roots and Radicals
y + 6 – 3 2y – 5 = 0Solve.
y + 6 = 3 2y – 5
y + 6 = 9 2y – 5 y + 6 = 18y – 45
6 = 17y – 45
51 = 17y
3 = y
3 + 6 – 3 2 • 3 – 5 ? 0
9 – 3 6 – 5 ? 0
0 = 0
3 – 3 1 ? 0
3 – 3 ? 0
Solve.
3 = y
Check:
y + 6 – 3 2y – 5 = 0