Lesson 4.4
Fractional Exponents and Radicals
Exercises (pages 227–228)
A
3. The denominator of the exponent is the index of the radical.
1
1
a) 16 2  16
4
b)
36 2  36
6
d)
32 5  5 32
2
f)
(1000) 3  3 1000
  10
1
1
c)
64 3  3 64
4
e)
(27) 3  3 27
 3
1
1
4. Write each decimal exponent as a fraction.
1
a) 0.5 
b)
2
0.25 
1
1
4
1
1000.5  100 2
810.25  814
 100
 10
So,
c)
 4 81
0.2 
3
So,
1
5
d)
0.2 
1
1
5
1
10240.2  1024 5
(32)0.2  (32) 5
 5 1024
 5 32
4
So,
 2
So,
1
5. Use the rule: x n 
n
x
1
3
1
a) 36  3 36
b)
48 2  48
1
c)
(30) 5  5 30
1
6. Use the rule:
n
x  xn
1
39  39 2
a)
1
b)
4
1
c)
3
29  29 3
90  90 4
1
d)
5
100  100 5
7. a) Any number raised to the exponent 0 is 1.
80  1
1
b) 8 3  3 8
2
 1
c) 8   8 3 
 
2
3

2
 8
3
2
 22
4
3
d) 8 3  81
8
 1
e) 8   8 3 
 
4
3

4
 8
3
4
 24
 16
f)
5
 1
83   83 
 

5
 8
3
 25
 32
5
B
m
 a  or
  4  or
8. Use a n 
2
a)
2
3
43



2.3
3
b)
 10 5
3
c)
m
n
2.32 
n
3
42
10
5

3
am .

3
or 5 (10)3
or 2.33
9. The formula for the volume, V, of a cube with edge length e units is:
V = e3
To determine the value of e, take the cube root of each side.
3
V  3 e3
3
V e
Substitute: V = 350
1
e=
3
350 or e = 350 3
1
3
The cube has edge length
m
 a  or a .
  48  or 48
10. Use a n 
2
a)
48 3
m
n
c)
 1.8 3
2.5 

m
2
3
5
b)
n

3
350 cm or 350 3 cm.
3
1.8

5
2
or 3 (1.8)5
5
2
 3
 
8
2.5
5
 3 2
 
8
5
5
 3
 3
 
 or  
8
 8
So,
d) 0.75 
3
4
3
0.750.75   0.75  4
So,


4
0.75

3
or 4 0.753
2
e)
2
2
 5 5  5 5 
 5
5
       or   
9
 9
 9

f) 1.5 =
3
2
1.5
1.25
 1.25

So,

3
2
1.25

3
or 1.253
m
n
11. Use:
am  a n
3
a)
3.83  3.8 2
3
Or, since  1.5,
2
b)

c)
9
 9 4
4
   
5
5
3
1.5

2
3.83  3.81.5
2
  1.5 3
5
5
Or, since
5
5
 1.25,
4
4
1.25
9
9
   
5
 
5
4
4
 3
 3 3
d) 3     
8
8
3
3
e)
 5
 5 2

   
4
 4
3
3
Or, since  1.5,
2
f)
5
 2.5
3
Or, since
1.5
 5
5

   
4
 4
3
  2.5 5
3
 0.6,
5
5
 2.5
3
  2.5
0.6
 1
12. a) 9   9 2 
 
3
2

3
 9
3
 33
 27
b)
2
1


 27  3  27  3 
     
 8 
 8 


 27 
  3

 8 
3
 
2
9

4
c)
 27 
2
3
2
2
1


  27  3 




3
27
  3
2

2
2
2
9
3
2
d) The exponent 1.5 =
3
So, 0.361.5  0.36 2
1


  0.36 2 




0.36
 0.63
 0.216
2
e)
 64  3
1


  64  3 




3
64
  4 
 16
2

2
2
3

3
f)
3
 4 2
  
 25 
1


2
 4  
 25  


 4 
 

 25 
2
 
5
8

125
3
3
3
1
is equivalent to taking the square root of the number.
2
1
So, to write an equivalent form of each number using a power with exponent , square the number,
2
then write it as a square root.
a) 22  4
13. Raising a number to the exponent
1
So, 2 = 4 2 , or 4
b) 42  16
1
So, 4 = 16 2 , or 16
c) 102  100
1
So, 10 = 100 2 , or 100
d) 32  9
1
So, 3 = 9 2 , or 9
e) 52  25
1
So, 5 = 25 2 , or 25
1
is equivalent to taking the cube root of the number.
3
1
So, to write an equivalent form of each number using a power with exponent , cube the number,
3
then write it as a cube root.
14. Raising a number to the exponent
a)
 1
3
 1
1
So, –1 =  1 3 , or
3
1
b) 23  8
1
So, 2 = 83 , or 3 8
c) 33  27
1
3
So, 3 = 27 , or 3 27
d)
 4
3
  64
1
So, –4 =  64  3 , or
e)
43  64
1
So, 4 = 64 3 , or 3 64
3
64
3
 1 2
15. Since 4 is a perfect square, and 4 and   involve the square root of 4, I will evaluate these
4
numbers without a calculator. I can evaluate 42 using mental math. Because 4 is not a perfect cube, I
will use a calculator to evaluate 3 4 .
Use a calculator: 3 4  1.5874...
3
2
 1
4   42 
 
3
2

3
 4
3
 23
8
42 = 16
3
 1 2
  
4
1


2
 1  
 4  


 1
 

 4
1
 
2
1

8
3
3
3
3
 1 2
So, from least to greatest, the numbers are:   ,
4
16. a) i) The exponent 1.5 =
3
So, 161.5  16 2
 1
  16 2 



3
 16 
 43
 64
3
3
2
3
3
4 , 4 2 , 42
3
4
ii) The exponent 0.75 =
3
So, 810.75  814
 1
  814 



3
 81 
4
3
 33
 27
4
5
iii) The exponent 0.8 =
So,  32 
0.8
4
  32  5
1


  32  5 




5
32
  2 

4
4
4
 16
1
2
iv) The exponent 0.5 =
1
So, 350.5  35 2
 35 Since 35 is not a perfect square, use a calculator.
 5.9160...
3
2
v) The exponent 1.5 =
3
So, 1.211.5  1.212
1


  1.212 




1.21
 1.13
 1.331
3

3
vi) The exponent 0.6 =
3
5
So,
3
 
4
3
0.6
 3 5
 
4
3
 3
  5 
 4
 0.8414...
Since
3
is not a perfect fifth power, use a calculator.
4
b) I was able to evaluate the powers in parts i, ii, iii, and v without a calculator. I can tell
before I evaluate because: in part i, 16 is a perfect square; in part ii, 81 is a perfect fourth power;
in part iii, –32 is a perfect fifth power; and in part v, 1.21 is a perfect square.
2
17. Use the formula: h  35d 3
h  35  3.2 
 35

3
Substitute: d = 3.2
2
3
3.2

2
Since 3.2 is not a perfect cube, use a calculator.
 76.0036...
A fir tree with base diameter 3.2 m is approximately 76 m tall.
18. The rule for a power with a rational exponent is:
m
xn 
 x
n
m
3
In the first line, the student should have written 1.96 2 as

1.96

3
The correct solution is:
3
1.96 2 =

1.96
 1.4 

3
3
 2.744
19. Use the formula: SA  0.096m0.7
Substitute: m = 40
SA  0.096  40 
Use a calculator.
= 1.2697…
The surface area of a child with mass 40 kg is approximately 1.3 m2.
0.7
n
20. a) Use the expression: 100  0.5 5
Substitute: n =
1
2
n
5
100  0.5   100  0.5
1
2
5
1 1

5
 100  0.5  2
1
 100  0.5 10
Use a calculator.
 93.3032...
After
1
h, approximately 93% of caffeine remains in your body.
2
1
Use a calculator to evaluate the expression: 100  0.87  2  93.2737...
1
h, approximately 93% of caffeine remains in your body.
2
Both expressions give the same result, to the nearest whole number.
After
n
b) Use the expression: 100  0.5 5
Substitute: n = 1.5
n
1.5
100  0.5  5  100  0.5  5
 100  0.5 
0.3
Use a calculator.
 81.2252...
After 1.5 h, approximately 81% of caffeine remains in your body.
n
c) Use the expression: 100  0.5 5
Equate the expression to 50, the percent of caffeine that remains.
n
100  0.5  5  50
n
0.5 5 
Solve for n. Divide both sides by 100.
50
100
n
0.5 5  0.5
n
0.5 5  0.51
Equate the exponents.
n
1
5
n5
So, after 5 h, 50% of caffeine remains in your body.
3
21. Use the formula: T  0.2 R 2
To calculate the period for Earth, substitute: R = 149
3
T  0.2 149  2
Use a calculator.
 363.7553...
It takes approximately 363.8 Earth days for Earth to orbit the sun.
To calculate the period for Mars, substitute: R = 228
3
T  0.2  228  2
Use a calculator.
 688.5449...
It takes approximately 688.5 Earth days for Mars to orbit the sun.
So, Mars has the longer period.
C
22. Karen is correct. You can only multiply a number by itself a whole number of times.