I
Powers and Exponents
I66
Square and Cube Roots
I72
Scientific Notation
I78
You can use the proplems and
the list of words that follow t c
see what you already know
1
about this chapter.The answers
to the orobiems are in Hot
Solutions at the back of the book'and the definitions of the
words are in HotWords a t the front of the book.You can
find out more about a particular problem or word by
rmat
do
ya
already
know?
Problem Set
Write each multiplication, using an exponent. 3.1
1.3x3x3x3x3
2. a X a X a
3.9x9x9
4. x x x x x x x x x x x x x x x
Evaluate each square. 3.1
5. 32
6. 72
7. 4*
8. 82
Evaluate each cube. 3-1
9. 33
10. 43
11. 63
12. 93
Evaluate each power of 10.3'1
13. io4
14. lo6
15. 10''
16. 10'
Evaluate each square root. 3*2
17. V%
18.
19.
a
20. a
?
Powers and Roots
I65
Estimate each square root between two consecutive numbers. 3.2
21.
22. vi5
23. .\/71
24. I
&
m
Estimate each square root to the nearest thousandth. 3.2
25.
26. fi
27. @.I
28. V%
m
Evaluate each cube root. 3.2
29. fi
30. %
31.
32. %%6
Write each number in scientific notation. 3.3
33. 36,000,000
34. 600,000
35. 80,900,000,000
36. 540
Write each number in standard form. 3.3
37. 5.7 x lo6
38. 1.998 X lo’
39. 7 x lo8
40. 7.34 x io5
.ds
area 3.1
base 3.1
cube 3.1
cube root 3.2
exponent 3.1
factor 3.1
perfect square 3.2
power 3.1
scientific notation
3.3
square 3*1
square root 3.2
volume 3.1
I 6 6 HorTopics
5l
Powers and Exponents
,as you know, is the shortcut for showing a
+
+
repeaied addition: 4 X 2 = 2 2 + 2 2. A shortcut for
showing the repeated multiplication 2 X 2 X 2 X 2 is to write
Z4. The 2 is the factor to be multiplied, called the base. The 4 is
the exponent, which tells you how many times the base is to be
multiplied. The expression can be read as “2 to the fourth
power.”When you write an exponent, it is written slightly
higher than the base and the size is usually a little smaller.
write the multiplication u A u A u A 8 X 8 X
an exponent.
Check that the same factor is being used in the
multiplication.
All the factors are 8.
Count the number of times 8 is being multiplied.
There are 6 factors of 8.
Vrite the multiplication using an exponent.
Since the factor 8 is being multiplied 6 times, write 86
8X8X8X8X8X8=t
It
Vrite each multiplication using an exponent.
1.
2.
3.
4.
3X3X3X3
7x 7x 7x 7
aXaXaX aX aXa
z x z x z x z x z
Powers and Roots
I67
Evaluating the Square of a Number
The square of a number means to apply the exponent 2 to a
base. The square of 3, then, is 3‘. To evaluate 3’, identify 3 as
the base and 2 as the exponent. Remember, the exponent tells
you how many times to use the base as a factor. So 3’ means to
use 3 as a factor 2 times:
32 = 3 x 3 = 9
The expression 3’ can be read as “3 to the second power.” It can
also be read as “3 squared.”
3
3
When a square is made from a segment whose length is 3, the
area of the square is 3 x 3 = 3‘ = 9.
Evaluate 8’.
Identify the base and the exponent.
The base is 8 and the expo
1
Check It Out
Evaluate each square.
5. 4’
6 . 8’
7. 3 squared
8. losquared
I 6 8 HotTopics
Evaluating the Cube of a Number
To make the cube of a number means to apply the exponent 3
to a base. The cube of 2, then, is Z3. Evaluating cubes is very
similar to evaluating squares. For example, if you wanted to
evaluate 23, notice that 2 is the base and 3 is the exponent.
Remember, the exponent tells you how many times to use the
base as a factor. So Z3 means to use 2 as a factor 3 times:
23 = 2 X 2 X 2 = 8
The expression 23 can be read as “2 to the third power.” It can
also be read as “2 cubed.”
When a cube has edges of length 2, the volume of the cube is
2 x 2 x 2 = 23 = 8.
Check It Out
Evaluate each cube.
io. io3
9.53
11. 8 cubed
12. 6cubed
1
?owers and Roots
I69
Our decimal system is based on 10. For each factor of 10, the
decimal point moves one place to the right.
2 A l
--t
21.1
1 9 . 0 5 4 1,905
XI0
7.
W
-t
U
XI00
70
XI0
When the decimal point is at the end of a number and the
number is multiplied by 10, a zero is added at the end of the
number.
Try to discover a pattern for the powers of 10.
Powers
102
As a Multiplication
lox
10
104
OI
xOI
xOI xOI
105
OI
xOI
xOI xOI xOI
108
Result
Number
of Zeros
100
2
10,000
4
100,000
10 x 10 x 10 x 10 x 10 x 10 x 10 x 10
100,000,000
8
Notice that the number of zeros after the 1 is the same as the
power of 10. This means that if you want to evaluate lo7, you
simply write a 1 followed by 7 zeros: 10,000,000.
(4Check I t Out
Evaluate each power of 10.
13. 10’
15. lo8
14. lo6
16. io3
I70 HotTopics
Usually you think that
a zero means “nothing.”
But when zeros are related
t o a power of IO, you can get some fairly
large numbers. A billion i s the name for I followed
by 9 zeros; a quintillion is the name for I followed
by 18 zeros. You can write out all the zeros o
mathematical shorthand for these numbers.
I billion = 1,000,000,000 o r I
uintillion = I,OOO,OOO,OOO,OOO,OOO,O
at name would you use for I followed by one
hundred zeros?According t o the story, when th
mathematician Edward Kasner asked his 9-year-
used for IO ’ O O today.
Suppose you could count at the rate of I nu
each second. If you started counting now an
0
Z
a
”
Powers and Roots
I7 I
Write each multiplication using an exponent.
1.8X8X8
2 . 3 x 3 x 3 x 3 x 3 x 3 x 3
3. Y X Y X Y X Y X Y X Y
4. n X n X n X n X n X n X n X n X n X n
5. 15 X 15
Evaluate each square.
6. 5’
7. 14’
8. 7’
9. 1 squared
10. 20 squared
Evaluate each cube.
11. 53
12. 93
13. 1i3
14. 3cubed
15. 8cubed
Evaluate each power of 10.
16. 10’
17. 1014
18. lo6
19. What is the area of a square whose sides have a length of 9?
A. 18
B. 36
c. 81
D. 729
20. What is the volume of a cube whose sides have a length of 5?
A. 60
B. 120
C. 125
D. 150
w-