11.1 and 11.2 radicals goal(s): 1.to find the square roots of perfect squares, perfect square...
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11.1 and 11.2 Radicals
• Goal(s):
1. To find the square roots of perfect squares, perfect square radicands and estimate the roots of irrational numbers
2. Determine whether a number is rational or irrational
3. Determine acceptable replacements for radicands
Square Roots• When we raise a number to the second
power, we have “squared” the number.• Sometimes we need to find the number
that was squared. That process is called “finding the square root of a number”.
• Every positive number has both positive and negative square roots.
an25 5 5d -Radical Sign
25 25
25 5
“Principal” Square Root is the positive 5.
Simplify: (Answer is only the principal root)
81 9
64 -8 525 Negative square Root
Because (9)2 = 81
36(6)2 -36(-6)2 -36
Not possible
Perfect Square Radicands
|| 2 xx
Simplify: 21x
|1| x
Simplify: 1682 xx
|4| x
24x
Simplify: 225x
|5| x
|x|5or
Simplify:2
4
1x
x2
1 1
2x
• Comes from the word “ratio”
• Any number that can be expressed as the ratio of two integers
Rational Numbers
71
7,yes
1.310
13,yes
0.33331
,3
yes
Irrational Numbers• Cannot be written as the ratio of two
integers.• Decimal never ends and does not repeat.• Examples of irrational numbers:
2
0.4545 25
Identify the number as irrational or rational
35 36The square roots of most whole numbers are irrational. Only the perfect squares (0, 1, 4, 9,
16, 25, 36, etc.) have rational square roots.
6
1
Identify the rational number:
. 48
. 49
. 50
. 51
A
B
C
D
49 7
Real Numbers: All the rational and all the Irrational numbers.
Real Numbers
Rational Numbers
Irrational Numbers
25 is a real number.N T- O
2
95
0
7
25 25
25 25
Approximate the value of 29
25 5
36 6
29 5.3
Approximate the value of 12
9 3
16 4
12 3.4
Approximate the value of 75
64 8
81 9
75 8.7
Approximate the value of 45
36 6
49 7
45 6.8
Find without a calculator: 225
100 10
400 20
211 121212 144
213 169
15
Find without a calculator: 289
100 10
400 20
213 217
17
Radical Expressions(an expression written under a radical)
18 x 92 x
2
72 xRadicand
(the expression written under the radical)
The radicand must be positive!
Evaluate the expression for x = 5.Is the result a real number?
4x
945
3
Evaluate the expression for x = 2.Is the result a real number?
6x
462
number reala not
Determine the values of x that make the expression a real number.
12 x012 x
2
1x
12 x
Determine the values of x that make the expression a real number.
32 x
032 x
onsManySoluti
32 x
Homework
• Work book Page 38 ( 2-32) even