Transcript Find square roots

Changing Bases
Base 10: example number 2120
10³
10²
10¹
2
1
2
10⁰
Implied base 10
0 ₁₀
10³∙2 + 10²∙1 + 10¹∙2 + 10⁰∙0 = 2120₁₀
Base 8: 4110₈
8³
8²
8¹
8⁰
4
1
1
0₈
Base 8
8³∙4 + 8²∙1 + 8¹∙1 + 8⁰∙0 = 2120₁₀
Hexadecimal Numbers
Hexadecimal numbers are interesting. There are 16 of them!
They look the same as the decimal numbers up to 9, but then there are the letters
("A',"B","C","D","E","F") in place of the decimal numbers 10 to 15.
So a single Hexadecimal digit can show 16 different values instead of the normal 10 like this:
Decimal:
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Hexadecimal: 0 1 2 3 4 5 6 7 8 9 A B C D E F
Problem Solving:
3, 2, 1, … lets go!
Express the base 4 number 321₄ as
a base ten number.
Answer:
57
Add:
23₄ + 54₈ = _______₁₀
(Base 10 number)
Answer:
55
Subtract:
123.11₄ - 15.23₆ =
______₁₀
(Base 10 number)
Answer:
15 ⁴³⁄₄₈
Express the base 10 number 493 as
a base two number.
Answer:
111101101₂
Add:
347.213₁₀ + 11.428₁₀ =
________₁₀
(Base 10 number)
Answer:
358.641
Add:
234 + 324 =
________4
(Base 4 number)
Answer:
1214
Add:
234 + 324 =
________10
(Base 10 number)
Answer:
1214
Factorials
Factorial symbol ! is a shorthand
notation for a special type of
multiplication.
N! is written as
N∙(N-1)∙(N-2)∙(N-3)∙ ….. ∙1
Note: 0! = 1
Example: 5! = 5∙4∙3∙2∙1
= 120
Problem Solving:
3, 2, 1, … lets go!
Solve:
6! = _____
Answer:
720
Solve:
5!
3!
Answer:
20
Solve:
5!
3!2!
Answer:
10
Squares
Positive Exponents
“Squared”: a² = a·a
example: 3² = 3·3
=9
0²=0
1²=1
2²=4
3²=9
4²=16
5²=25
6²=36
7²=49
8²=64
9²=81
10²=100
11²=121
12²=144
13²=169
15²=225
16²=256
20²=400
25²=625
What is the sum of the
first 9 perfect squares?
Answer:
1+4+9+16+25+36+49+64+81=
285
Shortcut:
Use this formula
n(n+1)(2n+1)
6
Shortcut:
Use this formula
9(9+1)(2∙9+1)
6
Answer: 285
Square Roots
9.1
Evaluating Roots
1. Find square roots.
2. Decide whether a given root is rational,
irrational, or not a real number.
3. Find decimal approximations for
irrational square roots.
4. Use the Pythagorean formula.
5. Use the distance formula.
6. Find cube, fourth, and other roots.
9.1.1: Find square roots.
• When squaring a number, multiply the number by itself. To
find the square root of a number, find a number that when
multiplied by itself, results in the given number. The number a is
called a square root of the number a 2.
Find square roots. (cont’d)
The positive or principal square root of a number is written with
the symbol
The symbol –
The symbol
.
is used for the negative square root of a number.
, is called a radical sign, always represents the
positive square root (except that 0  0 ). The number inside the
radical sign is called the radicand, and the entire expression—radical
sign and radicand—is called a radical.
Radical Sign
Radicand
a
Find square roots. (cont’d)
The statement 9  3 is incorrect. It says, in part, that a positive
number equals a negative number.
EXAMPLE 1
• Find all square roots of 64.
Solution:
Positive Square Root
64  8
Negative Square Root
 64  8
Finding All Square
Roots of a
Number
EXAMPLE 2:
Finding Square Roots
•Find each square root.
Solution:
169
 225
25
64
 13
 15

25
64
5

8
EXAMPLE 3:
Squaring Radical Expressions
•Find the square of each radical expression.
Solution:

17
 29
2x  3
2
 17 

2
  29


 17

2
2x  3
2
 29

2
 2x2  3
9.1.2: Deciding whether a given root is rational,
irrational, or not a real number.
All numbers with square roots that are rational are called
perfect squares.
Perfect Squares
Rational Square Roots
25
25  5
144
144  12
4
9
4 2

9 3
A number that is not a perfect square has a square root that is
irrational. Many square roots of integers are irrational.
Not every number has a real number square root. The
square of a real number can never be negative. Therefore, -36
is not a real number.
EXAMPLE 4:
Identifying Types of
Square Roots
•Tell whether each square root is rational,
irrational, or not a real number.
Solution:
27
irrational
2
36  6 rational
27
not a real number
Not all irrational numbers are square roots of integers. For
example  (approx. 3.14159) is a irrational number that is not an
square root of an integer.
9.1.3: Find decimal
approximations for irrational
square roots.
A calculator can be used to find a decimal
approximation even if a number is irrational.
Estimating can also be used to find a decimal
approximation for irrational square roots.
EXAMPLE 5:
Approximating Irrational
Square Roots
Find a decimal approximation for each square root.
Round answers to the nearest thousandth.
Solution:
190
 99
 13.784048
 13.784
 9.9498743
 9.950
9.1.4: Use the Pythagorean formula.
Many applications of square roots require the use of
the Pythagorean formula.
If c is the length of the hypotenuse of a right triangle,
and a and b are the lengths of the two legs, then
a 2  b2  c2 .
Be careful not to make the common mistake thinking that a2  b2
equals a  b .
What is a right triangle?
hypotenuse
leg
right angle
leg
It is a triangle which has an angle that is
90 degrees.
The two sides that make up the right angle
are called legs.
The side opposite the right angle is the
hypotenuse.
The Pythagorean Theorem
In a right triangle, if a and b are the
measures of the legs and c is the
hypotenuse, then
a2 + b2 = c2.
Note: The hypotenuse, c, is always the
longest side.
The Pythagorean Theorem
“For any right triangle,
the sum of the areas of
the two small squares is
equal to the area of the
larger.”
a2 + b2 = c2
Proof
Find the length of the
hypotenuse if
1. a = 12 and b2 = 16.
2
2
12 + 16 = c
144 + 256 = c2
400 = c2
Take the square root of both sides.
2
400  c
20 = c
Find the length of the hypotenuse if
2. a = 5 and b = 7.
5 2 + 7 2 = c2
25 + 49 = c2
74 = c2
Take the square root of both sides.
74  c
8.60 = c
2
Find the length of the hypotenuse
given a = 6 and b = 12
1.
2.
3.
4.
180
324
13.42
18
Find the length of the leg, to the
nearest hundredth, if
3. a = 4 and c = 10.
42 + b2 = 102
16 + b2 = 100
Solve for b.
16 - 16 + b2 = 100 - 16
b2 = 84
2
b  84
b = 9.17
Find the length of the leg, to the
nearest hundredth, if
4. c = 10 and b = 7.
a2 + 72 = 102
a2 + 49 = 100
Solve for a.
a2 = 100 - 49
a2 = 51
2
a  51
a = 7.14
Find the length of the missing side
given a = 4 and c = 5
1.
2.
3.
4.
1
3
6.4
9
5. The measures of three sides of a
triangle are given below. Determine
whether each triangle is a right triangle.
73 , 3, and 8
Which side is the biggest?
The square root of 73 (= 8.5)! This must be
the hypotenuse (c).
Plug your information into the Pythagorean
Theorem. It doesn’t matter which number
is a or b.
Sides: 73 , 3, and 8
32 + 82 = ( 73 ) 2
9 + 64 = 73
73 = 73
Since this is true, the triangle is a
right triangle!! If it was not true, it
would not be a right triangle.
Determine whether the triangle is a right
triangle given the sides 6, 9, and 45
1. Yes
2. No
3. Purple
EXAMPLE 6
Using the Pythagorean Formula
Find the length of the unknown side in each right triangle.
Give any decimal approximations to the nearest thousandth.
Solution:
a  7, b  24
7 2  242  c 2
49  576  c 2
c  625
a 2  132  152
c  15, b  13
11
8
?
 25
a 2  169  225
a  56
82  b 2  112
b  57
625  c 2
a 2  56
 7.483
64  b 2  121
 7.550
b 2  57
EXAMPLE 7
Using the Pythagorean Formula
to Solve an Application
A rectangle has dimensions of 5 ft by 12 ft. Find the length
of its diagonal.
12 ft
5 ft
Solution:
52  12 2  c 2
25  144  c
2
169  c 2
c  169
c  13ft
9.1.5: Use the distance formula.
The distance between the points
d
 x1, y1 and  x2 , y2 is
 x2  x1    y2  y1 
2
2
.
EXAMPLE 8
Using the Distance Formula
• Find the distance between  6,3 and  2, 4 .
Solution:
d
 2   6   4  3
d  42   7 
d  16  49
d  65
2
2
2
9.1.6: Find cube, fourth, and other roots.
• Finding the square root of a number is the inverse of
squaring a number. In a similar way, there are inverses to
finding the cube of a number or to finding the fourth or
greater power of a number.
• The nth root of a is written
In
n
n
a.
a , the number n is the index or order of the radical.
Index
Radical sign
n
Radicand
a
It can be helpful to complete and keep a list to refer to of third and
fourth powers from 1-10.
EXAMPLE 9
Finding Cube Roots
• Find each cube root.
•
Solution:
3
64
4
3
27
 3
3
512
EXAMPLE 10
Finding Other Roots
• Find each root.
Solution:
4
81
 4 81
3
 3
4
81
Not a real number.
5
243
3
5
243
 3
9.2
Evaluating Roots
1. Multiply square root radicals.
2. Simplify radicals by using the product rule.
3. Simplify radicals by using the quotient
rule.
4. Simplify radicals involving variables.
5. Simplify other roots.
9.2.1: Multiply square root
radicals.
• For nonnegative real numbers a and b,
a  b  a  b and
a  b  a  b.
• That is, the product of two square roots is the square root of
the product, and the square root of a product is the product
of the square roots.
It is important to note that the radicands not be negative numbers in the
product rule. Also, in general, x  y  x  y .
EXAMPLE 1
Using the Product Rule to
Multiply Radicals
•Find each product. Assume that x  0.
Solution:
3 5
 35
 15
6  11
 6 11
 66
13  x
 13  x
 13x
10  10
 10 10
 100
 10
9.2.2: Simplify radicals using the product rule.
• A square root radical is simplified when no
perfect square factor remains under the
radical sign.
• This can be accomplished by using the
product rule:
a b  a  b
EXAMPLE 2
Using the Product Rule to
Simplify Radicals
•Simplify each radical.
Solution:
60
 4  15
 2 15
500
 100  5
 10 5
17
It cannot be simplified further.
EXAMPLE 3
Multiplying and Simplifying
Radicals
•Find each product and simplify.
Solution:
10  50
6 2
 10  50
 500
 100  5
 6 2
 12
2 3
 10 5
9.2.3: Simplify radicals by using the quotient
rule.
• The quotient rule for radicals is similar to the
product
• rule.
EXAMPLE 4
Using the Quotient Rule to
Simply Radicals
•Simplify each radical.
Solution:
4
49

4
49

48
3

48
3
 16
5
36
5

36
5

6
2
7
4
EXAMPLE 5
Using the Quotient Rule to
Divide Radicals
• Simplify.
Solution:
8 50
4 5
8 50
 
4
5
50
 2
5
 2  10
 2 10
EXAMPLE 6
Using Both the Product
and Quotient Rules
• Simplify.
Solution:
3 7

8 2

3 7

8 2

21
16

21
16

21
4
9.2.4: Simplify radicals involving variables.
• Radicals can also involve variables.
• The square root of a squared number is
always nonnegative. The absolute value is used
to express this.
2
For any real number a,
a  a.
• The product and quotient rules apply when
variables appear under the radical sign, as long
as the variables represent only nonnegative
real numbers
x  0, x  x.
Simplifying Radicals Involving
Variables
EXAMPLE 7
•Simplify each radical. Assume that all
variables represent positive real numbers.
Solution:
x
6
x
3
100 p 8
 100  p 8
7
y4

7
y4
Since  x
 10 p 4
7
 2
y

3 2
 x6
9.2.5: Simplify other roots.
• To simplify cube roots, look for factors that are perfect
cubes. A perfect cube is a number with a rational cube root.
• For example, 3 64  4 , and because 4 is a rational number,
64 is a perfect cube.
• For all real number for which the indicated roots exist,
n
a  n b  n ab and
n
a na

b  0 .
n
b
b
EXAMPLE 8
Simplifying Other Roots
•Simplify each radical.
Solution:
3
108
 3 27  3 4
 33 4
4
160
 4 16 10
 4 16  4 10
4
16
625
4
16
4
625
2

5
 2 4 10
Simplify other roots. (cont’d)
• Other roots of radicals involving variables
can also be simplified. To simplify cube roots
with variables, use the fact that for any real
number a,
3
a3  a.
• This is true whether a is positive or negative.
Simplifying Cube Roots Involving
Variables
EXAMPLE 9
•Simplify each radical.
Solution:
3
z
9
 z3
3
8x 6
 3 8  3 x6
3
54t 5
 3 27t 3  2t 2
15
3
a
64
3
15
a
 3
64
 2x 2
 3 27t 3  3 2t 2
a5

4
 3t 3 2t 2
9.3
Adding and Subtracting Radicals
1. Add and subtract radicals.
2. Simplify radical sums and differences.
3. Simplify more complicated radical
expressions.
9.3.1: Add and subtract radicals.
• We add or subtract radicals by using the distributive
property. For example,
8 36 3
 86 3
 14 3.
Only like radicals—those which are multiples of the same
root of the same number—can be combined this way. The
preceding example shows like radicals. By contrast, examples of
unlike radicals are
2 5 and 2 3,
as well as 2 3 and 2 3 3.
Note that
5 + 3 5 cannot be simplified.
Radicands are different
Indexes are different
EXAMPLE 1
Adding and Subtracting
Like Radicals
• Add or subtract, as indicated.
8 52 5
3 11  12 11
7  10
Solution:
 8  2 5
  3  12  11
 10 5
 9 11
It cannot be
added by the
distributive
property.
9.3.2: Simplify radical sums and
differences.
• Sometimes, one or more radical expressions in
a sum or difference must be simplified. Then,
any like radicals that result can be added or
subtracted.
Adding and Subtracting Radicals
That Must Be Simplified
EXAMPLE 2
•Add or subtract, as indicated.
27  12
Solution:
3 32 3
5 3
2 3 54  4 3 2
5 200  6 18

 5
5
  
2   6 9  2 
100  2  6
100 
92
2

3

27  3 2  4 3 2


 2 3 3 2  4 3 2
 50 2 18 2
 63 2  43 2
 32 2
 10 3 2
9.3.3: Simplify more complicated radical
expressions.
• When simplifying more complicated radical
expressions, recall the rules for order of
operations.
A sum or difference of radicals can be simplified only if the
radicals are like radicals. Thus, 5  3 5  4 5, but 5  5 3
cannot be simplified further.
EXAMPLE 3A
Simplifying Radical Expressions
•Simplify each radical expression. Assume that
all variables represent nonnegative real
numbers.
7  21  2 27
6  3r  8r
Solution:
 7  21  2 27
 7 3  2 27
 147  2 27
 6  r  2 2r
 3 2r  2 2r
 7 3  2 3 3
 6  3r  2 2r
 5 2r
 49  3  2 27
 7 36 3
 18r  2 2r
 49  3  2 27
 13 3
 9  2r  2 2 r
 
EXAMPLE 3B
Simplifying Radical
Expressions (cont’d)
•Simplify each radical expression. Assume that
all variables represent nonnegative real
numbers.
2
3
3
4
4
y 72  18 y
Solution:
y
 

9 8 

 
9  2 y2
 y 3 8  3 2 y2
    3
 y 3 2 2

 
y 6 2  3

2 y2
2  y2



81x  5 24 x

 
 6 2y 3 2y

 3 2y
 3x  3 3x  5  2 x   3 3x
 3y 2
 3x  3 3x  10 x  3 3x
3
27 x3  3 3x  5

 13x 3 3x
 
3
8 x 3  3 3x

