Post on 01-Jan-2016
Students will be able to estimate a square root, simplify a square root, and add and
multiply square roots.
Every whole number has a square root
Most numbers are not perfect squares, and so their square roots are not whole numbers.
Most numbers that are not perfect squares have square roots that are irrational numbers
Irrational numbers can be represented by decimals that do not terminate and do not repeat
The decimal approximations of whole numbers can be determined using a calculator
2 x 2 = 4or
What is a perfect Square?
2
2
A perfect square is the number that represents the area of the square.
The perfect square is 4
€
22 = 4
5
5
5 x 5 = 25
The perfect square is 25.
OR
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52 = 25
The inverse of squaring a number is to take the square root of the number.Think of it as you are given the area of a square, how long is each side.The square root of 4 is
2
The square root of 16 is
4
Perfect Squares (Memorize)1
4
916
253649
64
81
100121
144169196
225
256
324
400
625
289
By definition 25 is the number you would multiply times itself to get 25 for an answer.
Because we are familiar with multiplication, we know that 25 = 5
Numbers like 25, which have whole numbers for their square roots, are called perfect squares
You need to memorize at least the first 15 perfect squares
Perfect square
Square root
1
4
9
16
25
36
49
64
81
100
121
144
169
196
225
Perfect square
Square root
€
1 =1
€
4 = 2
€
9 = 3
€
16 = 4
€
25 = 5
€
36 = 6
€
49 = 7
€
64 = 8
€
81 = 9
€
121 =11
€
100 =10
€
144 =12
€
169 =13
€
196 =14
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225 =15
Obj: To find the square root of a number
• Find the square roots of the given numbers• If the number is not a perfect square, use a calculator to find the answer correct to the nearest thousandth.
81
37
158
6.083
12.570
€
81 = 9
€
37
€
158
Obj: Estimate the square root of a number
• Find two consecutive whole numbers that the given square root is between
• Try to do this without using the table
18
115
18 is between 4 and 5
115 is between 10 and 11
16 = 4 and 25 = 5 so
100 = 10 and 121 = 11 so
Complete Text Book p 385A.
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t = 9.52 = 90.25
B.
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t = 8.52 = 72.25C.
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t = 7.62 = 57.76
D.
The tension increases as the wave speed increases
Text p 3861.
€
v 2 = 812.
€
v 2 = 36
The wave speed must be 9 because the square root of 81 is 9.The wave speed must be 6 because the square root of 36 is 6.
3.Yes -9 because (-9)(-9) is also 81.Yes -6 because (-6)(-6) is also 36
4.
€
4 =
− 25 =
− 100 =
49 =
2
-5
-10
7
Text p 388
10.
€
13
€
9 <13 <16
€
9 < 13 < 16
€
3 < 13 < 4
€
13 ≈ 3.6
11.
€
30
€
25 < 30 < 36
€
25 < 30 < 36
€
5 < 30 < 6
€
30 ≈ 5.5
12.
€
75
€
64 < 75 < 81
€
64 < 75 < 81
€
8 < 75 < 9
€
75 ≈ 8.7
Steps To Simplify Radicals
€
8
To SIMPLIFY means to find another expression with the same value. It does NOT mean to find the decimal approximation. Step 1: Find the LARGEST PERFECT SQUARE that will divide evenly into the number under the radical sign. That means when you divide, you get no remainders, no decimals, no fractions. Perfect square 4 8/4=2
Step 2: Write the number appearing under the radical sign as the product (multiplication) of the perfect square and your answer from dividing.
€
8 = 4 *2Step 3: Give each number in the product its own radical sign.
€
8 = 4 * 2Step 4: Reduce the “perfect” radical that you have now created.
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8 = 2 2
4
16
25
100
144
= 2
= 4
= 5= 10
= 12
8
20
32
75
40
= = =
=
=
2*4
5*4
2*16
3*25
10*4
= =
=
=
=
22
52
24
35
102
Perfect Square Factor * Other Factor
LE
AV
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AD
ICA
L F
OR
M
18
288
75
24
72
= = =
=
=
= =
=
=
=
Perfect Square Factor * Other Factor
LE
AV
E I
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AD
ICA
L F
OR
M
€
9 * 2
€
2 6
€
144 * 2
€
25 * 3
€
4 * 6
€
36 * 2
€
12 2
€
5 3€
3 2
€
6 2
Simplify
€
3 50
Don’t let the number in front of the radical distract you. It is simply “along for the ride” and will be multiplied by our final answer
€
3 50 = 3 25*2
€
3 25 * 2
€
3*5 2
€
15 2
€
3 48
€
5 80
50
€
7 125
450
= = =
=
=
€
3 16* 3
€
5 16*5
2*25
€
7 25*5
2*225
= =
=
=
=
€
12 3
€
20 5
225
€
35 5
215
Perfect Square Factor * Other Factor
LE
AV
E I
N R
AD
ICA
L F
OR
M
Multiplying Radicals
Step 1: Multiply the numbers under the radical and multiply the numbers outside the radical.
€
2 6 *5 8
€
10 48Step 2: Simplify if possible
€
10 16 * 3
€
10* 4 3
€
40 3
35*5 175 7*25 75
Multiply and then simplify
73*82 566 14*46
142*6 1412
204*52 10020 20010*20
Simplify the following expressions
49
764 + 9
-4
255 +
= -2
= 7 8 + 9
= 56 + 9 = 65
= 5 5 + 7
= 25 + 7 = 32
€
5( )2= 5*5 25 5
€
7( )2= 7*7 49 7
€
8( )2= 8*8 64 8
€
x( )2= xx * 2x x
€
14 54
€
2 9
To divide radicals: divide the coefficients, divide the radicands if possible, and rationalize the denominator so that no radical remains in the denominator
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=7 6
7
56 8 2*4 22