Roots and Powers
Written by: Bette Kreuz
Edited by: Science Learning Center Staff
The objectives for this module are to:
1. Raise exponential numbers to a power.
2. Extract the root of an exponential number.
a) Extract the root when the exponent is not evenly
divisible by the root.
b) Extract the root when the power has a
numerator ≠ 1.
Introduction
Frequently, in your science courses, you will be asked to
solve problems that involve raising a number to a power
(such as squaring or cubing) or extracting a root (such
as the square root or cube root) in order to obtain a
numerical answer. Also, in dealing with extremely large
or small numbers, it is helpful to work with the number
in exponential form.
For these reasons, it is often necessary to know how to
raise exponential numbers to a power & how to extract
the root of a number in exponential form.
PART 1 –
RAISING EXPONENTIAL
NUMBERS TO A POWER
When a number, say, 200, is squared, it is
written as 2002. The number 2, slightly above
and to the right of the number 200, is a “Power”.
From your previous math courses you know that:
2002 = 200 x 200 = 40,000
PART 1 –
RAISING EXPONENTIAL
NUMBERS TO A POWER
In general, when a number is raised to a power, it
means that the number is multiplied by itself the
number of times given by the power. For example,
two hundred cubed, or raised to a power of three, is
written:
2003
The numerical value of 2003 is:
200 x 200 x 200 = 8,000,000
PART 1 –
RAISING EXPONENTIAL
NUMBERS TO A POWER
Three additional examples are:
(0.20)2 = 0.20 x 0.20 = 0.040
(10)4 =10 x 10 x 10 x 10 = 10,000
(0.20)3 =0.20 x 0.20 x 0.20 = 0.0080
(Note that there are two significant figures in the
answer when there are two significant figures in the
number being raised to a power.)
PART 1 –
RAISING EXPONENTIAL
NUMBERS TO A POWER
An exponential number, for example 2.00 x 102,
can also be raised to a power. The terms "power"
and "exponent" can be used interchangeably for the
superscript on 10.
In this module, we will use the term “exponent”
when referring to the superscript on 10 and “power”
for the superscript on an exponential number:
(2.00 x 102)2 Power
Exponent
Study the examples below. Each example is
written in several different, yet equivalent,
exponential forms:
(2.00 x 102)2 = (2.00 x 102) x (2.00 x 102) =
4.00 x 104
(2.00 x 102)3 = (2.00 x 102) x (2.00 x 102) x
(2.00 x 102) = 8.00 x 106
(2.00 x 10-1)2 = (2.00 x 10-1) x (2.00 x 10-1) =
4.00 x 10-2
(2.00 x 10-1)3 = (2.00 x 10-1) x (2.00 x 10-1) x
(2.00 x 10-1) = 8.00 x 10-3
From studying the preceding examples you can see that
there are two rules for raising an exponential to a power.
These rules are:
1. (the value of the exponent of 10 in the answer) =
(the exponent of 10 in the original number) x
(the value of the power)
(2.00 x 102)2
=4.00 x 104
2x2=4
(2.00 x 102)3
=8.00 x 106
2x3=6
(2.00 x 10-1)2
=4.00 x 10-2
-1 x 2 = -2
(2.00 x 10-1)-3
=0.125 x 103
-1 x -3 = 3
*Note that it is necessary to carry through the appropriate sign into the answer.
2. (the value of the coefficient in the answer) =
(the coefficient of the original exponential
number) multiplied by itself the number of
times indicated by the power
(2.00 x 102)2
= 4.00 x 104
2.00 x 2.00 = 4.00
(2.00 x 102)3
= 8.00 x 106
2.00 x 2.00 x 2.00 = 8.00
(2.0 x 10-1)2
=4.0 x 10-2
2.0 x 2.0 = 4.0
(2.00 x 10-1)-3
= 0.125 x 103
hint: (2.00 ) = (
.)
×
×
=
. . . . PART 1 –
RAISING EXPONENTIAL
NUMBERS TO A POWER
The general formula for raising an
exponential number to a power is shown
below:
(10A)B = 10A x B
RULE 1
OR
(Z x 10A)B = (Z)B x 10 A x B
RULE 2
PART 1 –
RAISING EXPONENTIAL
NUMBERS TO A POWER
Below are two worked examples of exponential
numbers raised to a power.
(4.10 x 10-2)3 = (4.10)3 x 10-2x3 =
68.9 x 10-6 OR 6.89 x 10-5
(6.00 x 104)2 = (6.00)2 x 104x2 =
36.0 x 108 OR 3.60 x 109
PROBLEM SET I
Now it's your turn. On a separate piece of paper
work the problems below to obtain the correct
numerical answer. You may check your work
with the answers at the end of the module.
1) (3.00 x 10-3)4
2) (5.6 x 108)2
3) (1.00 x 10-4)6
PART 2a - EXTRACTING THE
ROOT OF AN EXPONENTIAL
NUMBER
To obtain a square root of a number, say, 400
(“ 400”), we try to find the number that when
multiplied by itself two times produces the
original number (20 x 20 = 400). This process is
just the opposite of raising a number to a power.
PART 2a - EXTRACTING THE
ROOT OF AN EXPONENTIAL
NUMBER
Look at the examples below:
(Remember that 41/2 = 40.5, 161/4 = 160.25, etc.)
400 = 20 OR (400)0.5 = 20 OR (400)1/2 = 20
SINCE 20 X 20 = 400
8,000,000 = 200 OR (8,000,000)1/3 = 200
SINCE 200 X 200 X 200 = 8,000,000
The operation of root extraction can be
designated either with a radical sign where n
indicates the root (
) or with a fractional
power, ⁄, where n again represents the root.
64 = 641/2 = 640.5 = 8
125= 1251/3 = 1250.33 = 5
81 = 811/4 = 810.25 = 3
Usually, in science courses, the fractional power
designation is used and this designation will be
used in the rest of this module.
FOR EXAMPLE:
10 = (103)1/3 = 101 = 10
10
= (1012)1/3 = 104 = 10,000
(10-16)0.25
= 10 = (10-16)1/4 = 10-4 = 0.0001
Study the two examples on the next slide &
you will notice that the same rules used to
raise exponential numbers to a power can be
used to extract a root from exponential
numbers .
EXAMPLE I:
(4.00 x 106)0.5 = (4.00 x 106)1/2 =(4.00)1/2 x 10(6 x ½)
= (4.00)1/2 x 106/2
= (4.00)1/2 x 103
= 2.00 x 103
EXAMPLE II:
(8.00 x 10-9)1/3 = (8.00)1/3 x 10-9 x 1/3
= (8.00)1/3 x 10-9/3
= (8.00)1/3 x 10-3
= 2.00 x 10-3
Rule 1. Multiply the fraction designating
the root by the power of ten to obtain the
power of ten in the final answer:
EXAMPLE I 6 x 1/2 = 3
EXAMPLE II -9 x 1/3 = -3
Rule 2. Take the appropriate root of the
coefficient:
EXAMPLE I (4.00)1/2 = 2.00
EXAMPLE II (8.00)1/3 = 2.00
Extracting roots poses a special problem when
the exponent is not evenly divisible by the root
given in the fraction.
For example,
(9.00 x 105)0.5 = (9.00 x 105)1/2 = (9.00)1/2 x 10(5 x ½)
= 3.00 x 105/2 OR 3.00 x 102.5
This answer (105/2 or 102.5) is not in standard
exponential form and is an awkward expression to
work with.
In such cases, it is necessary to change the
coefficient to a power of 10 that is evenly divisible
by the root we are extracting as is shown in the next
slide.
9.00 x 105 = 90.0 x 104
THEN: (9.00 x 105)1/2 = (90.0 x 104)1/2
= (90.0)1/2 x 104x½
= (90.0)1/2 x 104/2
= (90.0)1/2 x 102
= 9.49 x 102
Before we demonstrate additional examples, we
briefly review (next 3 slides) the rules about moving
the decimal point in the coefficient & changing the
power of 10 without changing the value of the
number…
These rules are:
1) If the decimal is moved to the right, subtract
from the exponent of 10 the number equal to
the number of places the decimal was moved.
(1.60 x 105)1/2 = (16.0 x 105-1)1/2 = (16.0 x 104)1/2
(4.700 x 107)1/4 = (4700 x 107-3)1/4 = (4700 x 104)1/4
2) If the decimal is moved to the left, add to the
exponent of 10 the number equal to the
number of places the decimal was moved.
(3.42 x 105)1/3 = (0.342 x 105+1)1/3 = (0.342 x 106)1/3
(1.00 x 108)1/3 = (0.100 x 108+1)1/3 = (0.100 x 109)1/3
Note that it really doesn't matter which way the decimal
point is moved. You will get the same answer either way.
For example:
= (2500 x 10-4)1/2
DECIMAL MOVED TO
RIGHT
(250.0 x 10-3)1/2
= .5000 x 100
= (25.00 x 10-2)1/2
DECIMAL MOVED TO
LEFT
OR
= (3789000 x 10-3)1/3
DECIMAL MOVED TO
THE RIGHT
(3789 x 100)1/3
[ Note: (100)1/3 = 100 = 1]
= 1.559 x 101
= (3.789 x 103)1/3
DECIMAL MOVED TO
THE LEFT
PART 2a –
EXTRACTING THE ROOT OF AN
EXPONENTIAL NUMBER
The general formula for extracting the root of
an exponential number is shown below:
(Z x 10A)1/N
RULE 1
(Z)1/N x 10A x 1/N
(Z)1/N x 10A/N
RULE 2
Make sure that the exponent of 10 is evenly
divisible by N. If it isn’t, change the exponent
by moving the decimal.
PROBLEM SET II
It’s your turn again. Work the problems and check
your work with the answers at the end of the
module.
1)
2)
3)
4)
(3.60 x 105)1/2 OR (3.60 x 105)0.5
(2.70 x 107)1/3
(1.00 x 1012)-1/4 OR (1.00 x 1012)-0.25
(1.60 x 105)1/4 OR (1.60 x 105)0.25
PART 2b –
EXTRACTING THE ROOT OF AN
EXPONENTIAL NUMBER
What do you do when you have powers where the
numerator is not 1?
FOR EXAMPLE:
(9.0 x 104)1.5 OR (9.0 x 104)3/2 = (9.0 x 104)3 x 1/2 =
(9.0 × 10 )
You can multiply the number in parentheses by
itself the number of times that the numerator
indicates, and then take the root that the
denominator indicates as you did using the rules
from the previous section.
(9.000 x 104)3/2 = (9.00 x 104)3 x 1/2 =
[(9.00 x 104) x (9.00 x 104) x (9.00 x 104)]1/2
=(729 x 1012)1/2
= (7.290 x 1014)1/2 = 2.700 x 107
To help the manipulation of the number and the
exponent to get an "easy" number to work with,
consider the following 2 ideas:
1. Try extracting the root first & then carry out
the power term indicated in the problem.
For example:
(4.0 x 100)3/2 = [(4.0 x 100)1/2]3
EXTRACT THE SQUARE ROOT
= [2.0 x 100]3
RAISE THE ANSWER TO THE 3rd POWER
= 8.0 x 100
2. In extracting a root consider moving the
decimal point to get an "easy" root OR
changing the exponent to get a number easily
divisible by the root being taken.
(Do whichever makes the math less complicated!)
Example 1:
= (64 x 102)1/2
(“Easy” root)
(6.4 x 103)1/2
8.0 x 101
= (0.64 x 104)1/2
(The square root is less
obvious here)
Example 2:
(3.43 x 104)1/3
= (34.3 x 103)1/3
= (0.0343 x 106)1/3
Both give an exponent evenly divisible by 3,
BUT, the root of one may be easier to estimate
than the other.
(34.3 x 103)1/3
(0.0343 x 106)1/3
The root is between 33 (27) and
43 (64)
The root is between 0.33 (0.027)
and 0.43 (0.064)
= 0.32 x 102
= 3.2 x 101
Both answers are the
same.
= 3.2 x 101
PROBLEM SET III
Using the “Easy” Root/Make the Math Less
Complicated approach, try the following
problems. Check your answers at the end of the
module.
1) (8.00 x 100)4/3
2) (3.0 x 10-1)4/2
CALCULATOR SUPPLEMENT:
If you are using a calculator with only a square root button on
it and run across a problem like the following:
(1.728 x 109)1/4 OR (1.728 x 109).25
It may be helpful to think of 1/4 as (1/2 x 1/2)
(1.728 x 109)1/2 x 1/2 OR ((1.728 x 109)1/2)1/2
and just take the square root of the number in parentheses
twice! (once for each time it is multiplied by the 1/2 power.)
((17.28 x 108)1/2)1/2 = (4.157 x 104)1/2 = 2.039 x 102
Notice that this will only work for numbers where the power
has a denominator that is a power of 2, such as -1/4, -1/2, 0,
1/2, 1/4, 1/8, 1/16, 1/32, etc.
PROBLEM SET IV
If you the Calculator Supplement was useful to you
because your calculator only has a square root button,
practice doing the following problems on your calculator:
1) (10-8)-1/4 OR (10-8)-.25
2) (409.6 x 101)1/4 OR (409.6 x 101).25
3) (656.1 x 101)1/8
4) (2.401 x 1011)1/2 OR (2.401 x 1011)0.5
5) (8.55 x 100)3/4 OR (8.55 x 100)0.75
PROBLEM SET I ANSWERS
1) 81.0 x 10-12 OR 8.10 x 10-11
2) 31 x 1016 OR 3.1 x 1017
3) 1.00 x 10-24
PROBLEM SET II ANSWERS
1)
2)
3)
4)
(36.0 x 104)1/2 = 6.00 x 102
(27.0 x 106)1/3 = 3.00 x 102
1.00 x 10-3
(16.0 x 104)1/4 = 2.00 x 101
PROBLEM SET III ANSWERS
1) 16.0 x 100
2) 9.0 x 10-2
PROBLEM SET IV ANSWERS
1)
2)
3)
4)
5)
1.0 x 102
8.000 x 100
3.000 x 100
4.900 x 105
5.00 x 100
Done!
If you have any questions or are having difficulty
solving these problems, ask the assistant in the
Science Learning Center for help. If you feel
competent in raising exponential numbers to a
power and extracting roots from exponential
numbers, ask the assistant for the post-test.