Download 2-6 Exponents

2-6-1 Introduction to Exponents
Repeated Multiplication
Remember multiplication is a way to write repeated addition. To say 3+3+3+3 we write 3x4.
Sometimes multiplication is done over and over and over. To write 3x3x3x3 we write 34.
42 (read “four squared”) means 4x4 so 42 =16
53 (read “five cubed”) is 5x5x5=125
Practice: Write the indicated operation using exponents. Then perform the operation.
a) 4x4x4=
10X10X10X10X10X10=
2x2x2x2x2x2x2x2=
b)
5x5=
3x3x3x3x3=
c)
0.3 x 0.3 x 0.3=
d)
0.2 x 0.2 x 0.2 x 0.2=
0.01x0.01x0.01x0.01=
e)
-3 x (-3) x (-3) x (-3)=
-100x(-100)x(-100)=
f)
aaaaa=
(answer: a5)
5.1 x 5.1=
xxxxxxxxxxx=
9x9x9=
(-1.2)(-1.2)(-1.2)=
1.5x1.5=
(-1)x(-1)x(-1)x(-1)x(-1)=
2f2f2f2f=
Practice: Perform the indicated operation.
g) 25=
72=
82=
h)
122=
152=
252=
i)
(-4)2=
(-10)2=
(-13)2=
j)
(-4)3=
(-12)2=
(-5)3=
k)
(-10)3=
(0.5)4=
(0.2)4=
1
Review Topic 7: Fraction Multiplication and Division
Look at the following examples. Decide what the rule is for multiplying fractions.
Examples: 2/3 x 1/5 = 2/15,
Rule:
3 3 9
5
5 3 15 7
  ,
 3     1 , 4/5 x 3/ 4 =12/20=3/5
8 1 8
8
4 4 16 8
Notice: Fraction numbers must be written as the ratio of two integers
to be multiplied.
a c ac
 
b d bd
Thus to multiply 2 ½ x 1 ¾ x 12
1. First change the mixed numbers to improper fractions 5/2 x 7/4 x12/1
2. to get 420/8
3. This can be simplified to 52 ½.
One other step can save time. Canceling reduces the answer before you multiply the numbers.
3
3
15 7 12 15
7 12
3  7  7 147
  
 


 14 7
10
2 4 25
2
4 25 5 2  1 5 10
1
Fraction Division
To divide fractions, invert the second fraction and multiply.
Rule:
a c a d ad
  x 
b d b c bc
2/3  ½ = 2/3 x 2/1 = 4/3 = 1 1/3.
Notice: DON”T cancel until after the divisor (second number) is inverted.
For word problems, be very careful which number to write first in the problem. The thing being split,
cut apart, or sorted goes first.
1
1 10 5 10 2 2 2 4 1
3  2        1
3
2 3 2 3 5 3 1 3 3
1.
2.
3.
4.
5.
Change to improper
Invert and Multiply
Cancel
Multiply
Simplify
Practice:
a) 1
2
=
5 3
1  2
 =
5  3
1 2
  =
5 3
1  2
  =
5  3
b)
1 2
 =
5 3
1  2
 =
5  3
1 2
  =
5 3
1  2
  =
5  3
c)
2
 
3

2
2
=
 2
 
 3
2
=
 2
 
 3
3
=
 2
 
 3
4
=
d)
1  3 3
5     1 =
4  7 5
e)
3
3
4 3 
8
4
f)
53=
g)
Jack has 2 ¾ sacks of flour. If he uses 2/3 of the
flour, how much flour is left?
Tiffany splits 4/5 pounds of nails into 4
piles. How many pounds in each pile?
h)
15 ½ pounds of chocolate is put in ¾ pound
boxes. How many full boxes are made? How
much chocolate is left?
A cube (box with all sides the same
length) is 3 ½ inches on each side.
What is the volume of the cube?
Hint: Multiply all three sides. (3 ½)3
i)
5 ¾ purdinkles are split into buckets that hold 2/3
of a purdinkle. How many buckets will we need?
(You don’t need to know what a purdinkle is.)
Find the surface area of the cube above.
Hint: 6 times (3 ½ )2
j)
2 ¾ yards of fabric are needed for each shirt.
How many yards for 10 shirts?
How many cubes can be made of a
piece of metal 367 ½ square inches?
How many shirts could be made from 30 yards?
How much metal is left over?
1  3  3
5       1  =
4  7  5
3
3
4  3 
8
4
3  3
4   3  
8  4
(-3)4=
(-2)7=
1  3  3
5       1  =
4  7  5
3  3
4   3  
8  4
(-8)2
k) Write a word problem that requires multiplication of fractions then subtraction to solve.
3
2-6-2 Word Problems
a) If one yard of polar fleece fabric costs
$12.00, how much does 5 ½ yards cost?
b) Janet worked with students for 5 ¾ hour
on Tuesday, 4 3/8 of an hours on
Wednesday, and 1 2/3 hours Thursday. How
much time did she spend with students for
the three days?
One cubic foot of water weighs 62 ½ pounds.
How much does 5 3/8 cubic feet of water
weigh?
My mileage gauge reads 4533 3/10 miles.
(Ok, I admit it has rolled over.)
Last week it read 4042 7/10 miles.
How far have I driven this week?
c) The shopping list included 3 pounds of
hamburger, 3 ½ pounds of apples, 2 ¾
pounds of oranges, and 1/3 pound kiwi.
How many pounds of fruit were purchased?
The baby was 6 2/5 pounds at birth
two months ago.
He gained 4 ½ pounds.
How much does he weigh now?
d) How many pieces of wire each 3 ¾
inches long can be cut from a roll that is 50
inches long?
One slide is 5 3/8 feet taller than the other.
The short slide is 12 ¾ feet.
How tall is the tall slide?
e) The records showed 123 ¾ pounds of
monkey food were still in the inventory.
Each day 25 2/3 pounds are used. How
much is left after three days?
Monster costumes for the play each required
5 3/5 yards of fabric. How many yards will be
needed to make all 12 costumes?
f) How many quart jars of jam can be made
from 34 5/8 pound of strawberries? 2/3 of a
pound will make one quart jar.
How many 1 2/3 pound boxes of cookies can be
made from 10 ¾ pounds of cookies?
g) Jamie spent 3 ¾ hours typing word
problems, 2 ½ hours making answer keys,
2/3 of an hour making copies. One twelfth of
her time was spent yelling at the computer.
The rest of her eight hour day was spent
filing. How much time was spent filing?
One section of a wall of my house will be a book
shelf. The section is 6 ½ feet long and 10 feet
high. Most shelves should be 1 foot tall, but the
two bottom shelves will be 1 ¾ feet tall for
oversized books. This will leave the top shelf an
odd height. How much shelving is needed?
Don’t put shelving on the ceiling, but the floor
will need shelving.
4
h) While doing home health visits, Jack
drove 3 ½ miles to Mrs. Jenkins, 5 7/10
miles to the next appointment, 4 2/5 to
Mr. Smith, and 7 miles back to the office.
What should he claim on his mileage report?
Jon will work overtime for time and a half. He
works a 10 hour day. His regular pay is 12.50
per hour. How much did he make?
i) The family got three 2 ½ pound boxes of
chocolate for Christmas. How much
chocolate does each of the five people get?
Larry was home from work because he was ill
for 5/12 of his 40 hour work week. How many
hours did he work?
j) Jon's full truck weighs 3 ¾ tons. The
average crate weight 2/5 ton. If there are
seven crates on the truck, how much does
the empty truck weigh?
If you change the rotating speed of a cement
mixer from 12 rpm's to 5 1/4 rpm's how many
revolutions per minute slower does the cement
mixer spin?
k) Jackie lost 10 3/5 lbs. Before the diet she
weighed 140 ½ lbs. How much does she
weigh now?
l) Zuella made some dresses out of 15 ¾
yards of fabric. Each required 3 2/3 yards.
How many dresses was she able to make?
m) A ten pound bag of cotton balls has been
opened. (It's a really big bag.) Yesterday
there was 8 2/3 pounds left. If someone
uses 7/8 of a pound, how much do the
remaining cotton balls weigh?
n) Craft trim is to be cut in 5 3/8 in. pieces.
How many pieces can be cut from 3 ½
yards?
Rosy cut 3 ¾ feet from a 10 foot piece of sheet
metal. How long was the remaining piece?
I only have ¾ pounds of German baking
chocolate left. My recipe calls for 5/8 pounds.
How much will I have left?
Todd worked 9 ½ hours Monday, 10 ¾ hours
on Tuesday, 8 hours on Wednesday, 8 hours on
Thursday, and 9 hours on Friday. He makes
$14 per hour regular pay and time and a half for
anything over 40 hours. How much did he make
that week?
The trip to the store is 7/8 miles. Joan walked
2/3 of the way. How far has she walked?
5
2-6-3 Exponent Rules
Remember:
34  3(3)(3)(3)
x5 x3  xxxxxxxx  x8
Rule: a a  a
m
n
so
34  33   3(3)(3)(3)(3)(3)(3)  37
23 s2 32 s5  8ss9sssss  72sssssss  72s 7
and
m n
so
f 52 f 123  f 175
1
7
3
and 2w 8 3w 4  6w 8
(don’t worry what fractional exponents mean…yet.)
3
5 5
35 15
8 6
When multiplying like bases, add the exponents. 3 y z 4 y z  12 y z  12 y z
Practice: Simplify.
a)
a1a1 =
c 2c7 =
e2e2 
b)
b 6b 3 
d 5d 1 =
c)
a 10 a 10 =
c 25 c 33 =
f 4 f 8=
e 71e 53 
d)
b 600b 30 
d 5 d 1 =
f
e)
4a 1 4a 1 =
8c 2 5c 7 =
2e 2 2e 2 
f)
4b 7 7b 4 
3d 5 9d 2 =
g)
25 y 3 32 a =
4 2 c 3 52 c 7 =
34 f 3 4 3 f 8 =
Go M Et s 
h)
36 b 6 b 3 
5 2 d 5 4d 1 =
43 f 4 63 f 8 =
i)
3a 1 x 2 a 2 x 1 3 =
4 x 3 c 5 x 9 10c 7 =
(9  1) xe4 x10 e 4 
j)
x 5 2b 5 2 x 3 b 3 
x 3.4 d 5 x 3 d 2 5 =
7x6 f 4 x9 f 9 =
Remember how to cancel. Make a 1 out of common factors.
N
f N=
g9g7 =
h5h5 =
g 29 g 47 =
6
14  6  7  2  7


4 15  2  5  5
r 5 rrrrr rrrrr rr
h3
hhh
1
1
2




r



r3
rrr
rrr
1
h6 hhhhhh hhh h3
a3b2 aaabb
aaabb
a
a




 2
2 4
a b aabbbb aabbbb
bb b

am
mn
Rule: n  a
a
k 543
 k 543 215  k 328
215
k
12s 25 3 9
 s
20s16 5
When dividing like bases, subtract the exponents.
5
Review the examples.
6
r
rrrrr rrrrr rr


  r2
3
r
rrr
rrr
1
is the same as
r 53
5
h 3h 2 =
Ng 9 Ng 7 =
3( 31) h 7 3 4 h 5 =
44 g 4 44 g 4 =
5 2 h 5 55 h 5 =
1.5 x1.1 g 9 7.5 x 2.1 g 6 =
4 x 6 h 7 x9 h 2 =
Practice: Simplify.
a) w4
b)
c)
d)
e)
e5

e3
t8

t3
x 54

x110
14 f 14

21 f 12
15 y13 21y 52
32 x 45

w2
q5

q4
q 24

q10
4e7

12e 2
2 x3 5 x5
15 x 4
Notice what happens with the example
y7

y5
x 21

x 21
w84

w92
144k 12

60k 5
12a 5 56a 81
64a 50
r8

r2
w4

w4
m 45

m 73
45 x 64

9 x 54
4a 25 5a 8
20a 30
p12

p8
b14

b13
f 31

f
72h 46

56h 45
3x3 2 x 2

5 x5
h3
hhh
1
1



 h36  h3
6
3
h
hhhhhh hhh h
This leads to another rule.
s 3 
1
s3
and
1
 d5
5
d
Rule: a
n

1
1
and
 an
n
n
a
a
Practice: Write each answer two ways, as a fraction
with positive exponents and without using fractions but using negative exponents.
f)
w4
x
w32
1
-3



w
or
5
7
34
3
g)
w
a7

a12
w
x
q 41

q 52
36r 2r 5 s12 9  4  2rr 5 s12 8r 6 s12 8 67 127

 7 7  r s
45r 7 s 7
9  5r 7 s 7
5r s
5
h)
i)
3e3
4e 6
30 j 30
25 j 60
j)
28 x 5 y 5
21x 3 y 7
k)
15m3 3m3
45m6
w
s4

s5
=
8s 5
5r
=
w4

w2
z14

z15
8 1 5
r s
5
7f7
4f6
j3
5 j10
9 p3
3 p5
12t 8
15t 15
36 y13
6 y16
56r 98
48r 123
81m3n5
144m3n7
35s 2t 5
21s 3t 17
32 x 51 y15
12 x 30 y 71
25m5 33m5
55m10
6 p1310 p13
8 p16
15t 3 60t 3
45t 7
7
l)
32t 16
56t 25
m)
10k 12l 51
45k 36l 27
n)
3m14 3m13
45m61
x3
 x 3 3  x 0
3
x
2t
6t
42e 65
48e 79
k 31l
5k 36l 7
30k l 52
10k 36l 3
 
Rule: a
m
n
a
mn
Practice: Simplify.
q)
2 3
r)
s)
t)
u)
v)
8

x  
2x  
 4a b c 
2
3
3
5
  2 2 
  
 5  
Rule: a  1
0
(4y)0=
2s5x0=
x 
3 5
 64a 6b15c 3
2
=
 3xy 3 
 6 x5 y 2 


2
=
(3st5)0=
2(3st5)0
 x3 x 3 x 3 x 3 x 3  x 35  x15
(5-8+9-7-12)0=
½ (3st5)0
using a previous rule.
An easier way would be to multiply the 3 and 5 to get the new exponent
of 15.
x 
12 3
y 
 5r 
21 3
2
24 f 12e41
4e 26 f 27
or all cancel for an answer of 1.
Anything to the zero power is 1.
Practice: Simplify
o) 30=
x0=
0
p) 4x =
8e0=
s 
12t 90
48t 205
25
4
5
4

w 

m

 3t 
4 5
 3d e f  =
 2.5 

2 3
 x5 y 3 
 x5 y 2 


3

3
3
=


u 
 3x 
52 6

2

2
5
3 5

 2u v u 
5
12
 4a

 0.2 x 0.5 x 
5
=
x 
2
2 5
4
5 5
2

3 2
 56kl 310k 3 
 16k 8l 214k 10 



2
=
21
2b51 
2
 0.5r 1.2r 
3
4 3
 12a 3 4a 5b3 
 6 a 8b 5 


2
=
5x3  3x2 are not like terms and cannot be combined.
3
4
7
Contrast this with 5x 3x  15x The multiplication can be done.
4m3  5m3  9m3 Don’t add the exponents. 4m3 5m3  20m6 Do add the exponents.
a)
3d 5  2d 5 
3d 5 2d 5 
2 x3  2 x5 
Notice:
b)
7e5 2e5 
7e5  2e5 
3u3  2u5 
c)
4s-7s+8s2=
4s(-7s)(8s2)=
7r4+7r3=
d)
3x2-(-4x2)-8x2=
3x2(-4x2)(-8x2)=
w-6w-(-7w2)=
Mixed practice: Simplify
e) 4q3 p18 5q 8 p 9 =
f)
 2 3 a  3 4 a  512 a
g)
 36 x8 y 3 
 48 x 7 y12 


2 x3 2 x5 
3u3 2u5 
7r4(7r3)=
w(-6w)(-7w2)=
2  3t 3   5t 2 
4q3 p18  5q8 p 9  4q3 p18 =
 3nm 
2
2
 5nm 2 2nm 2 =
3a 2  1 
3 a 
5  3 
4
=
4
3
4e2  7e3  3e =
 2w 
3 2
 7 w3w5
34w3

h)
3x 2 4 x 2  7 x 2 =
 4e  7e  3e  
 3s  5s
i)
2
y6 z 67 y3

5 4
12
15 y z
5 x 2 x 4 y5 x 3

5
10 xy 
7h 5 18h9

2 
3 5  h2 
j)
3
2
5
8
t 2  38 t 3  2 3 t 3 

3
8
5 3
3
t 2   38 t 3   2 3 t 3  

3
4

x3   x9 
3
10
9
2-6-4 Scientific Notation
Scientific notation is used to write very large and very small numbers.
The distance to the sun is150, 000,000km or 1.5x108km.
The bacteria Streptococcus pyogenes is 0.0000625 cm or 6.25x10-5cm long.
4x103=4x10x10x10=4x1000=4000 Notice: the exponent on the 10 is 3 and the decimal after the 4 moved to the
right 3 spaces.
2.3x105 = 23000 moving the decimal right 5 spaces.
2.3x105 is scientific notation and 23000 is decimal notation.
Practice: Write the following in decimal notation.
a) 4.6x108=
b)
7x103=
c)
1.123x109=
5.2x103=
3.4x1012=
8X108=
1x1010=
5.453x104=
6.5874x103=
Rule: a  10 is scientific notation if 1  a  10 and m is an integer.
m
2.3x108 = 23x107 = 0.23x109 = 230x106 But only one of these is in scientific notation.
2.3 is between 1 and 10 so the correct form of the number in scientific notation is 2.3x108.
Although the other numbers are the same as 2.3x108, they are not scientific notation.
3.21x102 is the scientific form of 321.
To write 45600000 in scientific notation, 4.56x107 = 45600000
move the decimal so only one digit is in front and drop ending zeros. 4.56
Multiply by ten to the power that matches the number of spaces the decimal was moved.
Practice: Write the following in scientific notation.
a) 8,900,000,000,000=
2,000,000,000,000,000=
b)
1,000,000,000,000=
c)
8,431,000,000,000,000=
Recall 10 
4
24,000,000,000,000=
1,000,000,000,000,000,000,000,000=
178,000,000,000,000=
145,000,000,000,000,000=
5,000,000,000,000=
1
1
6.8
4
so 6.8  10 
= 0.00068 Dividing by 10000 moves the decimal

4
10 10000
10000
4 spaces to the left.
To change a small number in scientific notation to decimal notation, move the decimal to the left the
number of spaces indicated by the power of 10.
5.43x10-3 = 0.00543 (left) and 6.43x103= 6430 (right)
Practice: Write the following in decimal notation. Watch the signs.
a) 4.6x10-8=
5.2x10-3=
b)
10
7x10-3=
1.8X10-8=
3.4x10-12=
1x10-10=
c)
1.123x10-9=
5.453x10-5=
6.5874x10-3=
d)
4.8x103=
2.3x10-5=
4.5x10-8=
e)
3x10-5=
3.3x10-6=
8.1x10-9=
To write a small decimal notation number to scientific notation, move the decimal after the first nonzero
digit. Count how many spaces the decimal was moved. This is the power of 10 the number would be
divided by. Make it a negative exponent.
0.00003123 = 0.00003 123  3.123  10
5
Practice: Write the following in scientific notation. Watch the signs.
a) 0.000000043=
0.00000254=
b) 0.0000235=
0.00007=
c) 0.000000000028=
0.00000000000009=
Practice: Write the missing numbers.
Decimal Notation
d)
e)
The population
6230,000,000.
of
the
Scientific Notation
The population of the United States is 3x108.
world
is
The diameter of an artery is .00621 inches.
The moon is 3.8x105 km away.
h)
i)
about
The human heart is 3.1x100 inches in diameter.
f)
g)
0.000023=
0.0000000001=
0.00001=
The mass of the moon is
73,500,000,000,000,000,000,000.
j)
The mass of the Earth is 5.97x1024 kg.
k)
A hair grows 2.08x10-4 mm per minute.
11
Multiplication and Scientific Notation
To multiply 3.1 x 104 X 2.3x105, use the commutative property 3.1x2.3x104x105 and the associative
property (grouping of multiplication doesn’t matter) to get 7.13x104x105.
Then use the exponent rules to get the final result 7.13x109.
(3x108)(1.5x1023)=4.5x1031
(2.1x104) (4x10-12) =8.4x10-8
(4.6x108) (5.8x106) =26.68x1014
Notice: What is wrong with the last example? Although the answer is correct, the number is not in
scientific notation. To finish the problem, move the decimal one space left and increase the exponent by
one.
26.68x1014 = 2.668x1015
If the decimal moves right, the exponent moves down.  0.00042x109=4.2x105
If the decimal moves left, the exponent moves up. 
7890x103=7.89x106
Practice: Multiply then write the following in scientific notation.
a)
3 105 2.3 109 
2.1105 1.3 109




b)
1.2 10 5.3 10  
c)
 2 10 3 10  
15
15
9

 2 10  
10
 7.2 10 110  
3.4 10  2.3 10  
3 10  2.3 10  
23
51
The negative in front of the 2 makes a
number less than zero. The negative in
front of the exponent makes a small
number.
9
5
5
9
9
 2.5 10 3 10  
5
12
The following will need to have the decimal moved and the exponent adjusted after the multiplication.
d)
5.4 107 3.1108 
3.25 109 5.3 1012 
2 1012 7.3 1023

e)
f)
12


5.48 10 3.2 10  
15
10



 7.2 10  4.3 10  
21
35



 2.5 10  8.3 10  
4
8
The population of the world is 6.23x109 and the average person consumes 5x103 grams of sugar per year. How
many grams of sugar are consumed in the world per year?
Division and Scientific Notation
4.5  108 4.5

 1085  1.5  103
Division works in a similar way.
5
3  10
3
1.092  1013 1.092

 1013( 5)  0.52  108  5.2  109 Notice the decimal and exponent.
5
2.1  10
2.1
6.8  103
3
8
 6.8  10    3.4  10   3.4  108  2  1038  2  105
Practice: Divide the following then write the answer in scientific notation.
a) 1.5  1021
5  1071
b)
c)

2.3  105
1103   1108  

2.3  1052
1103   3.4 108  
8.3  107

1.2  105
 6.8 103   3.4 108  
A certain string of bacteria is 3.4x10-5 cm long. There are approximately 80 cells in a string How wide is
each single bacteria?
Calculator notation
Some calculators show scientific notation the same way we have written them here.
Some use EE. They write 3.456 EE5 to mean 3.456 x105.
Some write the same thing with this in the screen 3.45605.
There are several ways electronic equipment displays scientific notation. Look in your user’s manual to
see how it is displayed on your calculator.
13
2-6-5 Roots
9  3 because 3 times 3 is 9
25  5 because 5 times 5 is 25
When working a square root problem, ask:
“What times itself is the number inside the root symbol?”
27  3 because 3x3x3 is 27. The small 3 outside the root symbol tells how many times the answer must be
multiplied to get the number inside the root.
3
5
32  2 because 2x2x2x2x2 or 25=32
Practice: Simplify.
a) 3 64 
b)
4
81 
c)
4
625 
d)
3
8
125 
5
1024 
5
243 
4
16 
3
3
216
64 
5
100000 
4
256 
6
Prime factorization is writing a number using multiplication of only prime
numbers.
12 can be written as 3x4 ,but 4 is not prime and can be written as 2x2,
so the prime factorization of 12 = 3x2x2 This can be written 3x22.
To write 330 using its prime factorization, start breaking it up into smaller
factors until there are no more composite numbers.
330=33x10=3x11x2x5
Prime number: a
number with only
factor of 1 and itself.
2 3, 5, 7,11,13,17,
19, 23... are prime
numbers.
15 is not prime
because 3 and 5 also
divide it evenly.
15 is a composite
number.
330
10
x
33
5 x 2 x 3 x 11
Practice: Write the prime factorization for the following.
a) 12
8
27
100
b)
56
28
36
32
c)
50
42
18
25
d)
75
48
360
72
e)
7875
50176
7200
3136
f)
405
432
5488
9375
g)
864
442368
3600
1225
14
Simplifying roots. You won’t be using the root button on your calculator for these.
45  3  3  5
 32  5  3 5
4
First write the number as it’s prime factorization.
Then, because this was a square root, a pair of 3s can be simplified to a 3 outside the root.
To get an exact answer, leave the 5 inside the root rather than using a calculator.
19440 Prime factorization
so
4
4
2  2  2  2  3  3  3  3  3  5 four 2’s and four 3’s and fourth root
2 4  4 34  4 3  5  2  34 15  64 15 This is the simplest form of the root.
Practice: Write the roots in simplest form.
a)
75  5  5  3  5 3
245 
18 
28 
b)
300  3  2  2  5  5  2  5 3  10 3
360 
1125 
392 
c)
576 
968 
2016 
3528 
3600 
70000 
2700 
d)
1575 
e)
3
500  3 5  5  5  2  2  53 4
3
1080 
3
576 
3
2744 
f)
4
400  4 2  2  2  2  5  5  24 25
4
567 
4
20000 
4
1296 
Using a calculator will give an approximation.
In the example above 4 19440  11.81 rounded to the nearest hundredth.
To use your calculator, you need to learn another notation.
Rule:
x
x
1
1
Example: 4 52  52 4  520 25 On your calculator find the xy button or yx.
Type 52 then hit the yx button. Then type .25 = 2.69 (Rounded)
Practice: First, rewrite the root using a fractional exponent. Then use your calculator to find the answer rounded to
the nearest hundredth.
10
5
g) 4 123  1231 / 4 type 123 yx then .25 =
100000 
159 
h)
3
25  25
i)
4
53  5 4  5.75 
j)
3
20 
1
3
type 25 yx then (1/3)=
4
578 
3
52  5
48 
2
3

3
951 
3
85 
345 
All the rules that work for exponents work for these fractional exponents. This is learned and practiced in future
math classes.
15