Math 2201
Unit 4: Radicals
Read Learning Goals, p. 173 text.
Ch. 4 Notes
§4.1 Mixed and Entire Radicals (1 class)
Read Goal p. 176 text.
Outcomes:
1. Define and give an example of a radical. See notes
2. Identify the index and the radicand of a radical. See notes
3. Define and give an example of an entire radical. pp. 175, 515
4. Define and give an example of a mixed radical. pp. 175, 516
5. Define and give an example of a perfect square. pp. 175, 517
6. Define and give an example of a principal square root. pp. 176, 517
7. Define and give an example of a secondary square root. pp. 176, 517
8. Simplify a simple radical. p. 178.
9. Convert an entire radical to a mixed radical. p. 178
10. Convert a mixed radical to an entire radical. p. 179
Def n : A radical is any expression that can be written in the form
n is called the index of the radical, and
b is called the radicand.
E.g.: 4  2 4 (index is 2, radicand is 4)
E.g.: 3 16 (index is 3, radicand is 16)
E.g.:
5
E.g.:
7
32 

 index is 5, radicand is

35 

1 12 
1 
x  index is 7, radicand is x12 
2
2 

32
35
n
b where:
E.g.: 7  2 7 (index is 2, radicand is 7)
E.g.: 4 100 (index is 4, radicand is 100)
E.g.:
x  4 (index is 2, radicand is x  4 )
Def n : A perfect square is a whole number that is the square of another whole number.
E.g.: If we square 6, we get 36. So 36 is a perfect square.
E.g.: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225 are perfect squares.
Finding the Square Root of a Number
Recall that numbers usually have two square roots, one positive and one negative.
E.g.: The square roots of 16 are 4 and -4 because  4   16 and  4   16 .
2
2
Def n : The principal square root of a number is the positive square root.
E.g.: 4 is the principal square root of 16 because 16  4 .
Def n : The secondary square root of a number is the negative square root.
E.g.: -4 is the principal square root of 16 because 16  4 .
Note that the principal square root is most commonly used in the real world.
Do # 1 (omit c), p. 182 text in your homework booklet.
Simplifying Simple Radicals
E.g.: Simplify 144
144  12
Why isn’t 144  12 ?
E.g.: Simplify
3
4
38416
38416  14
Why isn’t
4
38416  14
E.g.: Simplify
5
27
27  3
E.g.: Simplify
4
3
5
32768
32768  8
Def n : Entire radicals are radicals with a coefficient of 1.
56  1 56 is an entire radical.
E.g.:


E.g.:
3
108  1
E.g.:
4
 1

1 2
x  8  1 4 x 2  8  is an entire radical.
2
 2

3
108 is an entire radical.
Def n : Mixed radicals are radicals where the coefficient is NOT 1. They are written in the form k n b
where k is NOT 1.
E.g.: 2 5 is a mixed radical because the coefficient is 2.
E.g.: 5 7 is a mixed radical because the coefficient is -5.
E.g.: 
5
5
4a is a mixed radical because the coefficient is  .
7
7
Converting Mixed Radicals to Entire Radicals
You may be asked to convert a mixed radical into an entire radical. To do this, you should make use of
inverse operations like squaring and square root, cubing and cube root, and so on. For example,
7  72  49,  6  3  6   3 216, 11 
3
4
11
4
 4 14641,  5 
5
 5
5
 5 3125
E.g.: Convert 5 2 to an entire radical.
5 2  52 2  25 2 
 25 2  
50
E.g.: Convert 4 3 2 to an entire radical.
4 3 2  3 43 3 2  3 64 3 2  3 64  2  3 128
E.g.: Convert 3 4 6 to an entire radical.
3 4 6  4 34 4 6  4 81 4 6  4 81 6  4 486
E.g.: Convert 8 5 2 to an entire radical.
8 5 2  5  8
5 5
2  5 32768 5 2  5 32768  2  5 65536
Comparing and Ordering Radicals
To help order and compare radicals, it is often useful to write all radicals as entire radicals and compare
the radicands.
1
E.g.: Order from least to greatest: 6  7  2 , 7 2, 20, 20 3, 401
First convert each mixed radical to an entire radical.
1
6  7  2  6 7  36 7 
7 2  49 2 
 36  7  
 49  2  
252
98
20  400
20 3  400 3 
 400  3 
1200
401
The entire radicals from least to greatest are
98, 252, 400, 401, 1200 , so the original radicals
1
2
from least to greatest are 7 2, 6  7  , 20, 401, 20 3
Note that you could have also changed each radical to a decimal using a calculator and used the decimal
approximations to order the radicals.
Do # 11, p. 183 text in your homework booklet.
Converting an Entire Radical to a Mixed Radical
To convert entire radicals to mixed radicals, you often use perfect squares like 4, 9, 16, 25, 36, 49, 64,
81, 100, 121, 144, …, perfect cubes like 8, 27, 64, 125, …, and so on.
E.g.: Convert 128 to a mixed radical.
Find the largest perfect square that divides evenly into 128.
128  64 2  8 2
E.g.: Convert
3
250 to a mixed radical.
Find the largest perfect cube that divides evenly into 250.
3
250  3 125 2   3 125 3 2  5 3 2
Do #’s 2, 3 a, 4 a, c, 5, p. 182 text in your homework booklet.
§4.2 Adding and Subtracting Radicals (2 classes)
Read Goal p. 184 text.
Outcomes:
1. Define and give an example of like radicals. See notes
2. Add and subtract like radicals. pp. 184-187
3. Simplify radical expressions. pp. 184-187
Def n : Like radicals are radicals with the same index and the same radicand.
E.g.: 2 4 and  5 4 are like radicals because both have an index of 2 and a radicand of 4.
E.g.: 2 3 7 and  3 3 7 are like radicals because both have an index of 3 and a radicand of 7.
E.g.: 12 4 6 and 5 4 6 are like radicals because both have an index of 4 and a radicand of 6.
E.g.: 3 5 11 and 7 5 11 are like radicals because both have an index of 5 and a radicand of 11.
E.g.: 2 4 and  5 3 4 are NOT like radicals. Why not?
E.g.: 
157 8
3
x and 3 5 x8 are NOT like radicals. Why not?
2 4
4
E.g.: 
1 11 3 7
3
x and 311 x8 are NOT like radicals. Why not?
2 4
4
Do # 1, p. 188 text in your homework booklet.
Adding and Subtracting Radicals
Adding and subtracting radicals is the same as adding and subtracting monomials. Only like radicals can
be added or subtracted, just as only like terms can be added or subtracted with monomials.
E.g.: Simplify 2 x  5x  8x
Following the order of operations gives 2x  5x  8x  7 x  8x  1x   x
E.g.: Simplify 2 6  5 6  8 6
Following the order of operations gives 2 6  5 6  8 6  7 6  8 6  1 6   6
You can check your answer using a calculator.
Sometimes you may want to first simplify some or all of the radicals and then combine like radicals.
E.g.: Simplify 3 8  5 72  8 18
Simplifying the radicals and combining like radicals gives
3 4 2  5 36 2  8 9 2

 
 
 3 2 2 5 6 2 8 3 2

 6 2  30 2  24 2
 36 2  24 2
 12 2
You can check your answer using a calculator.
E.g.: Simplify  75  4 7  252  48
Simplifying the radicals and combining like radicals gives
 25 3  4 7  36 7  16 3
 5 3  4 7  6 7  4 3
 9 3  2 7
You can check your answer using a calculator.
E.g.: Simplify
13
2
3
4
32  3 135  3 500  3 1715
2
3
5
7
Simplifying the radicals and combining like radicals gives
13 3
2
3
4
8 4  3 27 3 5  3 125 3 4  3 343 3 5
2
3
5
7
1
2
3
4
 2 3 4  33 5  53 4  7 3 5
2
3
5
7
3
3
3
3
 4 2 5 3 4  4 5

 
 
 

 2 3 4  2 3 5
You can check your answer using a calculator.
Do #’s 2 b, d, 4, 5 b, d, 6 b, d, 9 a, 11, 14-16, 18, 19, pp. 188-190 text in your homework booklet.
§4.3 Multiplying and Dividing Radicals (3 classes)
Read Goal p. 191 text.
Outcomes:
1. Multiply radicals. pp. 192-196
2. Divide radicals. pp. 192-196
3. Explain what is meant by rationalizing the denominator. pp. 195, 517
4. Simplify radical expressions. pp. 192-196
Use a calculator to complete the following table.
n
an b
n
5 3 
5  3  15 
11  4 
11 4  44 
3
4 3 8 
3
4  8  3 32 
4
2  4 10 
4
2 10  4 20 
5
65 7 
5
6  7  5 42 
Make a conjecture about the value of
n
a  n b and the value of
n
ab
ab .
Conjecture: They are ________________________.
Multiplying Radical Expressions
When multiplying radical expressions, we multiply the coefficients and then multiply the radicands. To
multiply radicals, they must have the same index.


E.g.: Simplify 2 8 6 108

Since each radical has an index of 2, we have the choice of multiplying the radicals as they appear and
then simplifying or simplifying each radical first and then multiplying. We’ll do this example both ways.
Method 1: Multiply first, simplify next.
 2 8  6

  2  6   8 
108
108

 12 864
 12 144  6 
 12 144 6


 12 12 6  144 6
Method 2: Simplify first, multiply next.
 2 8  6 108 
  2 4 2  6 36 3 
 2  2 2   6  6 3  
  4 2  36 3 
  4  36   2  3   144
6
Do #’s 1, 4, p. 198 text in your homework booklet.

E.g.: Simplify 6 5 7 3  8 5

Since each radical has an index of 2, we can multiply these radicals using the distributive property.


  6 5  7 3    6 5 8 5 
  6  7   5  3    6  8   5  5 
6 5 7 3 8 5
 42 15  48 25
 42 15  48  5 
 42 15  240
Your Turn

Simplify 2 3 7 27  5

ANS: 126  2 15
Do #’s 5 a-d, p. 198 text in your homework booklet.


E.g.: Simplify 5 3  8 9 7  7 21

Since each radical has an index of 2, we can multiply these radicals using FOIL, or the box method, or
the happy rainbow method, etc.
5 3  89 7  7 21 
  5 3  9 7    5 3  7 21    8   9 7    8   7 21 
 45 21  35 63  72 7  56 21
 11 21  35 9 7  72 7
 11 21  35  3 7  72 7
 11 21  105 7  72 7
 11 21  33 7


E.g.: Simplify 3 8  4 2  7 3

Since each radical has an index of 2, we can multiply these radicals using FOIL, or the box method, or
the happy rainbow method, etc.
3 8  4 2  7 3 
  3 8   2    3 8  7 3    4  2    4   7 3 
 6 8  21 24  8  28 3
 6 4 2  21 4 6  8  28 3
 6  2  2  21 2  6  8  28 3
 12 2  42 6  8  28 3
E.g.: Simplify


20  24 3 12  5 32

Sometimes it may be better to simplify the radicals before multiplying. Since each radical has an index
of 2, we can multiply these radicals using FOIL, or the box method, or the happy rainbow method, etc.
Sometimes it may be better to simplify the radicals before multiplying.


20  24 3 12  5 32

 4 5  4 6 3 4 3  5 16 2 
  2 5  2 6  3  2  3  5  4  2 
  2 5  2 6  6 3  20 2 
  2 5  6 3    2 5  20 2    2 6  6 3    2 6  20 2 

 12 15  40 10  12 18  40 12
 12 15  40 10  12 9 2  40 4 3
 12 15  40 10  12  3 2  40  2  3
 12 15  40 10  36 2  80 3
Do #’s 5 e, p. 198 text in your homework booklet.
Sample Exam Question
Express
a)
b)
c)
d)

3 2

2
in simplest form.
1
52 6
1 2 3
5

E.g.: Simplify 5 3  8 2

2
Since each radical has an index of 2, we can multiply these radicals using FOIL, or the box method, or
the happy rainbow method, etc.
5 3  8 2 5 3  8 2 
  5 3  5 3    5 3  8 2   8 2  5 3   8 2 8 2 
 25 9  40 6  40 6  64 4
 25 9  80 6  64 4
 25  3  80 6  64  2 
 75  80 6  128
 203  80 6
Your Turn
Simplify

2 6

2
ANS: 8  4 3
Do #’s 5 f, 11, p. 198 text in your homework booklet.
Use a calculator to complete the following table.
n
a
n
b
15
 2.2361...
3
15
 5  2.2361...
3
22
 1.4142...
11
22
 2  1.4142...
11
3
8

4
3
8 3
 2
4
10

2
4
10 4
 5
2
64
2
2
5
64 5
 32  2
2
3
4
n
4
5
5
n
Make a conjecture about the value of
n
a
and the value of
b
Conjecture: They are ________________________.
n
a
.
b
a
b
Dividing Radical Expressions
When dividing radical expressions, we divide the coefficients and then divide the radicands. To divide
radicals, they must have the same index.
E.g.: Simplify
8 25
2 4
Since each radical has an index of 2, we can divide these radicals.
8 25
8 25 8 25  8  5 
40


       10
4
2 4 2 4 2 4  2  2 
E.g.: Simplify
64 56
16 7
Since each radical has an index of 2, we can divide these radicals.
 56 
64 56  64   56 

  4 
  4
 
16 7
 16   7 
 7 
E.g.: Simplify
 8   4  4  2   4  2  2   8
108 3 18
12 3 9
Since each radical has an index of 3, we can divide these radicals.
 18 
108 3 18  108   3 18 
3

  9 2
  3   9  3
3
12
9
12 9
9





E.g.: Simplify
98 4 56
49 4 8
Since each radical has an index of 4, we can divide these radicals.
 56 
98 4 56  98   4 56 
4


2


 4
  2 7


8
49 4 8  49   4 8 


E.g.: Simplify
115 3072
55 5 3
Since each radical has an index of 5, we can divide these radicals.
2
115 3072  11   5 3072   1   5 3072   1 

    
   

3   5 
55 5 3
 55   5 3   5  

5

4
 1
1024      4   
5
 5
Rationalizing the Denominator with One Term in the Denominator
By convention, when we simplify an expression containing radicals, we do not leave a radical in the
denominator. Instead, we change the irrational number to a rational number. This is called rationalizing
the denominator.
E.g.: Simplify
4
9 3
We want an equivalent fraction that does NOT have the radical in the denominator. To do this we will
multiply both the numerator and the denominator by 3 .
 4  3  4 3 4 3 4 3





 9 3   3  9 9 9  3 27
You can check by changing both
E.g.: Simplify
4
9 3
and
4 3
to a decimal approximation.
27
2
5
We have to change the irrational number in the denominator into a rational number.
2  2  5 



5  5   5 
2 5
 5  5 

2 5 2 5

5
25
We changed the irrational denominator

2  1.414213562
 into a rational number (5).
5 2
7
5 2  5 2  7  5 2 7 5 14 5 14




 
7
7  7 
7 7
49
 7 
E.g.: Simplify
E.g.: Simplify
2  2  


8  8  
2
8
8  2 8 2 4 2 2  2  2 4 2 1 2  2






8
8
2
2
8 
8 8
64
36 18
8 8
36 18  36 18  8  36 18 8 36 18  8 36 144 36 12 
 




 

8 8
8 8
8 8 8
8 88
8 64
 8 8  8 
432
216
108
54
27



 
64
32
16
8
4
E.g.: Simplify
Do #’s 2, 13, 14, 16 a, 19 a, 21 b, pp. 198-200 text in your homework booklet.
3 2 5
2
3 2  5  3 2  5  2  3 2 2  5 2 3 4  5 2 3  2   5 2 6  5 2
 




 
2
2
2
2 
2 2
4

 2 
E.g.: Simplify
Alternative Solution
We could have broken the expression into two fractions first and then simplified.
3 2  5 3 2 5  3 2  2  5  2  3 2 2 5 2
3 4 5 2


 




 

 

2
2
2  2  2 
2 2
2 2
2 2
4
4
3 2 5 2 6 5 2 6  5 2


 

2
2
2
2
2
E.g.: Simplify
6 2 3
3
6 2  3  6 2  3  3  6 2 3  3 3 6 6  9 6 6  3
 



 3  
3
3
3
3 3
9



Alternative Solution
6 2 3 6 2
3  6 2  3 
6 2 3
6 6
6 6
6 6 3


 
1 
1 
1 


  1 

3
3
3
3
3
3  3  3 
3 3
9

6 6 3
3
Do # 16 b, c, d, p. 199 text in your homework booklet.
Do #’s 2, b, f, 3 b, d, f, 4, 6 b, d, e, 7, b, e, 8 b, c, e, 10 a, c, e, 11, 12 a, c, d, p. 203 text in your home
work booklet.
§4.4 Simplifying Algebraic Expressions Involving Radicals (2 classes)
Read Goal p. 204 text.
Outcomes:
1. Simplify radical expressions containing variables. pp. 206-210
2. Identify the restrictions on the variable(s) in a radical expression. pp. 204, 517
Up to now, all our expressions involving radicals contained only numbers. Now we have to apply what
we have learned to radical expressions with variables.
Def n : The restrictions on an expression are the values for which the expression is defined.
1
1
is x  2 because
is defined for any real number except for 2.
x2
x2
E.g.: The restriction on x is x  0 because x is NOT defined for numbers less than 0.
E.g.: The restriction on
E.g.: The restriction on
x  6 is x  6 because
x  6 is NOT defined for numbers less than 6.
E.g.: The restriction on
x  2 is x  2 because
x  2 is NOT defined for numbers less than -2.
E.g.: There are no restrictions on
x 2 because
x 2 is defined for all real numbers.
E.g.: There are no restrictions on
x 4 because
x 4 is defined for all real numbers.
E.g.: There are no restrictions on
3
x because
3
x is defined for all real numbers.
E.g.: The restriction on
x3 is x  0 because
x3 is NOT defined for numbers less than 0.
E.g.: The restriction on
x5 is x  0 because
x5 is NOT defined for numbers less than 0.
Do #’s 1, 11, pp. 211, 213 text in your home work booklet.
Simplifying Radicals with Variables
Keep in mind that when variables are involved you must indicate the restrictions on the variable. To
determine the restrictions you must look at the original expression.
E.g.: Simplify 6 24x3
6 24 x3  6 24 x3  6 4 6 x 2 x  6  2  6  x  x  12 6 x x ; x  0
E.g.: Simplify 8z 3 96 z 5
8 z 3 96 z 5  8 z 3 96 z 5  8 z 3 16 6 z 4 z  8 z 3 4 6 z 2 z   8  4   z 3  z 2  6 z
 32 z 5 6 z  32 z 5 6 z ; z  0
E.g.: Simplify
4 y  32
4 y  32  4  y  8  4 y  8  2 y  8;
y 8
Do # 2, p. 211 text in your home work booklet.
Adding and Subtracting Radical Expressions with Variables
Adding and subtracting radicals with variables is the same as adding and subtracting radicals without
variables. Only like radicals can be added or subtracted, just as only like terms can be added or
subtracted with monomials.
E.g.: Simplify 3 x  5 x
Since 3 x and 5 x are like radicals (same index, same radicand) they can be combined.
3 x  5 x  8 x; x  0
E.g.: Simplify 4 16 x 4  32 x 4
4 16 x4  32 x 4  4 16 x 4  16 2 x 4  4  4  x 2  4 2 x 2  16 x 2  4 2 x 2 ; x 
E.g.: Fill in the missing steps in the simplification below.
9z z  4 z3
Step 1: 9 z z  4
z
Step 2: 9 z z  4
z
Step 3:
Step 4:
  
z

z
Do #’s 3 a, b, 5, 6a, 9 b, p. 212 text in your home work booklet.
Multiplying Radical Expressions with Variables

E.g.: Simplify 7 8 z 2
7
8z 2

  5z 3z 
  5z 3z   7
8 z2

  5z
 
 

 7  2  2  z  5 z 3 z  14 2  z  5 z 3
  5z

z   14  5  z  z   3  2  z 
3 z  7 4 2 z2
3 z
 70 z 2 6 z  70 z 2 6 z

E.g.: Simplify 5 x 3 6  4 x

Using the distributive property we get

 

 



5 x 3 6  4 x  5 x 3 6  5 x  4 x   15 6 x  20 x x  15 6 x  20 x x

E.g.: Simplify 9 y  5 3  7 y
Using FOIL, or the box method, or the happy rainbow method, etc. we get
9

 





y  5 3  7 y  9 y  3  9 y 7 y   5  3   5  7 y

 27 y  63 y 2  15  35 y  27 y  63 y  15  35 y  63 y  8 y  15


E.g.: Simplify 3 y  11
2
Using FOIL, or the box method, or the happy rainbow method, etc. we get
 3 y  11   3 y  11 3 y  11
  3 y  3 y    3 y  11  11  3 y   1111
2
 9 y 2  33 y  33 y  121
 9 y  66 y  121
Do #’s 3 c, 4 a, b, 6 b, 8 c, d, 9, c, d, p. 212 text in your home work booklet.
Dividing Radical Expressions with Variables
Don’t forget about rationalizing the deniminator and giving the restrictions.
E.g.: Simplify
18 z 4
6 z3

18 z 4
6 z3
18 z 2
6 z2 z

18 z 2  18   z 2 
z 3z z
 z  3z




 3 z ; z  0
  3 


z
6z z  6   z z 
z
z
 z
E.g.: Simplify
15 7  10 54 y 3
5 y
Breaking the expression into two fractions gives
15 7 10 54 y 3  3 7  y   2 9 6 y 3   y 





 y 
 y  
 y 
5 y
5 y
y


 


 3 7  y   2  3 6 y 2 y   y 




 y 
 y  
  y 
y


 

 3 7  y   6 6  y  y   y 


 
 y 
 y  
 y 
y


 



3 7 y 6 y 6 y 3 7 y  6 6 y2


;
y
y
y
y0
Do #’s 12, 4 c, d, 6 c, d, 10, 15, pp. 212-213 text in your home work booklet.
§4.5 & 4.6 Exploring and Solving Radical Equations (2 classes)
Read Goal pp. 214, 216 text.
Outcomes:
1. Identify strategies to solve radical equations involving square roots and cube roots. p. 215
2. Solve radical equations involving square roots and cube roots. pp. 216-221
3. Identify extraneous solutions. P. 217
Solving radical equations makes use of inverse operations.




Addition and subtraction are inverse operaions.
Multiplication and division are inverse operaions.
Squaring and taking the square root are inverse operations.
Cubing and taking the cube root are inverse operations.
Solving radical equations with square roots or cube roots often involves squaring or cubing both sides of
the equation. This process may introduce lead to solutions that do not work when substituted back into
the original equation. These solutions are called extraneous solutions. Therefore, when you solve by
squaring or cubing you have to verify each solution to ensure it is not extraneous. Extraneous roots can
also be identified from the restrictions on the variable(s)
E.g.: Solve
5y  8
Squaring both sides gives

5y

2
5 y  64
64
y ;
5
 82
y0
Since we squared both sides we must check for extraneous roots (verify the solution).
LHS 
 5   645  
So the solution is y 
E.g.: Solve
64  8  RHS
64
.
5
x  2  9
Squaring both sides gives

x2

2
  9 
2
x  2  81
x  83; x  2
Since we squared both sides we must check for extraneous roots (verify the solution).
LHS 
83  2  81  9  RHS
So 83 is NOT a solution. It is an extraneous root. There is no solution to this equation. Do you see why?
x  5  6  18
E.g.: Solve
x  5  6  6  18  6
x  5  12

x5

2
 122
x  5  144
x  149; x  5
Since we squared both sides we must check for extraneous roots (verify the solution).
LHS 
149  5  6  144  6  12  6  18  RHS
So the solution is x  149 .
E.g.: Solve
3
3z  6
Cubing both sides gives

3
3z

3
 63
3 z  216
3 z 216

3
3
z  72; z 
Since we cubed both sides we must check for extraneous roots (verify the solution).
LHS  3 3  72   3 216  6  RHS
So the solution is z  72 .
3
E.g.: Solve
7 x  34  3  2
3
7 x  34  3  3  2  3
3
7 x  34  5

3
7 x  34

3
 53
7 x  34  125
7 x  34  34  125  34
7 x  91
7 x 91

7
7
x  13
Since we cubed both sides we must check for extraneous roots (verify the solution).
LHS  3 7 13  34  3  3 91  34  3  3 125  3  5  3  2  RHS
So the solution is x  13 .
Do #’s 2 c, d, 5, 6, 8 b, c, 15, pp. 222-224 text in your home work booklet.
Do #’s 1 b, d, 2 b, d, 3, 4, c, d, 5 c, d, 7, 9 b, d, 10, b, d, 11, p. 228 text in your home work booklet.
Problem Solving with Radicals
L
where T is the time (in
9.8
seconds) for one complete swing (over and back) and L is the length of the rope (in metres). If it takes a
pendulum 3.0s to make one complete swing, find the length of the rope.
E.g.: The motion of a pendulum can be modeled using the formula T  2
3.0  2
L
9.8

L 
 3.0    2

9.8 

L
9.0  4 2
9.8
9.0  9.8   4 2 L
2
9.0  9.8  4 2 L

4 2
4 2
2.2m  L
2
The rope is about 2.2m long.
E.g.: A sphere has a surface area of 50.27m2 . Find the radius of the sphere. S. A.  4 r 2
50.27  4 r 2
50.27 4 r 2

4
4
50.27
 r2
4
r  2.0m
The radius of the sphere is about 2.0m.
4
E.g.: A sphere has a volume of 523.60m3 . Find the radius of the sphere. V   r 3
3
4
523.60   r 3
3
4

3  523.60   3   r 3 
3

3
1570.80  4 r
1570.80 4 r 3

4
4
1570.80
 r3
4
1570.80 3 3
3
 r
4
r  5.0m
The radius of the sphere is about 5.0m.
Do # 13, p. 223 text in your home work booklet.