Numeracy Centre for Teaching and Learning | Academic Practice | Academic Skills | Digital Resources Page 1 +61 2 6626 9262 | [email protected]| www.scu.edu.au/teachinglearning [last edited on 7 September 2017] Introduction to Exponents When children study mathematics in primary school, they learn that repetitive addition, such as 3 + 3 + 3 + 3 + 3, is the same as the multiplication 5 x 3. There are occasions when repetitive multiplication occurs. To write this in a simpler way, the concept of exponents was developed. A study of exponents also helps student have a greater understanding of the number system used. A very large number which fascinates mathematicians is the Googol. The Googol is written as: 10 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 This is equivalent to 10 10 10 ............... 10 10 for 100 terms × × × × × . Using exponents, a short-hand way of writing this is 100 10 . While this number is beyond the comprehension of most people, other very large numbers such as the number of atoms in a quantity of material or the distance across the universe are still extremely large and very difficult to write in standard form. The same could be said extremely small numbers such as the diameter of an atom. An understanding of exponents gives us a more convenient way of writing both very large and very small numbers. The first part of this module gives a thorough understanding of exponents leading to Scientific Notation which is used to represent numbers that are very large or extremely small. Examples of repetitive multiplication are: 3 3 3 3 3 × × × × is written in exponential form as 5 3 4 4 4 4 4 4 4 4 4 × × × × × × × × is written in exponential form as 9 4 x x x x x x × × × × × is written in exponential form as 6 x 2 2 2 a a a × × is written in exponential form as ( ) 3 2a It is important that brackets are used in this example to show that both the 2 and the a are being repeated three times.
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Numeracy
Centre for Teaching and Learning | Academic Practice | Academic Skills | Digital Resources Page 1 +61 2 6626 9262 | [email protected] | www.scu.edu.au/teachinglearning [last edited on 7 September 2017]
Introduction to Exponents When children study mathematics in primary school, they learn that repetitive addition, such as 3 + 3 + 3 + 3 + 3, is the same as the multiplication 5 x 3. There are occasions when repetitive multiplication occurs. To write this in a simpler way, the concept of exponents was developed. A study of exponents also helps student have a greater understanding of the number system used.
A very large number which fascinates mathematicians is the Googol. The Googol is written as:
This is equivalent to 10 10 10 ............... 10 10 for 100 terms× × × × × .
Using exponents, a short-hand way of writing this is 10010 . While this number is beyond the comprehension of most people, other very large numbers such as the number of atoms in a quantity of material or the distance across the universe are still extremely large and very difficult to write in standard form. The same could be said extremely small numbers such as the diameter of an atom. An understanding of exponents gives us a more convenient way of writing both very large and very small numbers. The first part of this module gives a thorough understanding of exponents leading to Scientific Notation which is used to represent numbers that are very large or extremely small. Examples of repetitive multiplication are:
3 3 3 3 3× × × × is written in exponential form as 53
4 4 4 4 4 4 4 4 4× × × × × × × × is written in exponential form as 94
x x x x x x× × × × × is written in exponential form as 6x
2 2 2a a a× × is written in exponential form as ( )32a
It is important that brackets are used in this example to show that both the 2 and the a are being repeated three times.
7 7 7 7 7− − − − −× × × × is written in exponential form as ( )57−
The Language of Exponents The power na can be written in expanded form as: [ ]............ for factorsna a a a a a a n= × × × × × The power na consists of a base a and an exponent (or index) n.
na
Saying Powers 53 can be said as ‘3 to the power of 5’, ‘3 to the 5th power’, ‘3 to the 5th’ or ‘3 raised to 5’. 94 can be said as ‘4 to the power of 9’, ‘4 to the 9th power’, ‘4 to the 9th’ or ‘4 raised to 9’.
With common exponents such as 2 and 3, special language is used.
28 is most commonly said as 8 squared. 35 is most commonly said as 5 cubed.
( )32a is said as ‘2a all cubed’ to emphasize that both the 2 and the a are being cubed. Powers containing numbers can be evaluated.
53 3 3 3 3 3 243= × × × × =
Many students will perform the multiplication 3 5 15× = which is incorrect.
When solving 64− , use the order of operation rules where the powers are performed before the multiplication.
6 64 1 4 1 4096 4096− − − −= × = × =
( )42 2 2 2 2 16− − − − −= × × × =
4 42 1 2 1 16 16− − − −= × = × =
Powers containing variable bases can be expanded
4x x x x x= × × ×
3ab a b b b= × × ×
4( )ab ab ab ab ab a a a a b b b b= × × × = × × × × × × × One last thing to note: when a variable does not appear to have an exponent, the exponent is 1.
1a a=
1x x=
( )1de de= or 1 1d e
Page 4
Common Powers and Roots
Squares Square Roots
Cubes Cube Roots Other Powers
Other Roots
21 1= 1 1= 31 1= 3 1 1=
41 1= 4 1 1= 22 4= 4 2=
32 8= 3 8 2= 42 16= 4 16 2=
23 9= 9 3= 33 27= 3 27 3=
43 81= 4 81 3= 24 16= 16 4=
34 64= 3 64 4= 44 256= 4 256 4=
25 25= 25 5= 35 125= 3 125 5=
45 625= 4 625 5= 26 36= 36 6=
36 216= 3 216 6=
27 49= 49 7=
51 1= 5 1 1=
28 64= 64 8= 52 32= 5 32 2=
29 81= 81 9= 53 243= 5 243 3=
210 100= 100 10=
Calculators Some calculators contain a ^ to calculate powers. For example: 2.35^4=30.49800625 Other calculators use a w key For example: 2.35w4=30.49800625 Other calculators use a ^ key For example: 2.35 ^ 4=30.49800625
Numeracy
Centre for Teaching and Learning | Academic Practice | Academic Skills | Digital Resources Page 1 +61 2 6626 9262 | [email protected] | www.scu.edu.au/teachinglearning [last edited on 7 September 2017]
Module contents Introduction
• Exponent Rules – Whole Number Exponents • Exponent Rules – Integer and Fraction Exponents • Equations with Exponents • Scientific Notation • Operations with Numbers in Scientific Notation
Answers to activity questions Outcomes
• Name the parts of a number written in exponential form. • Simplify exponents using rules. • Solve equations containing exponents. • Express numbers in scientific notation. • Perform operations with numbers in scientific notations.
Check your skills This module covers the following concepts, if you can successfully answer these questions, you do not need to do this module. Check your answers from the answer section at the end of the module. 1. (a) Simplify:
6 34× ×x x x (b) Simplify:
6 0÷a a (c) Simplify:
( ) ( )2 43 4 24 2× ÷x x x 2. (a) Simplify:
( )41
22 02− × ×
x x x
(b) Simplify: 3 512
(c) Simplify:
( )2
9 6 327x y
3. (a) Solve for x: 4 81=x
(b) Solve for x: 43 1 7+ =x
(c) Solve for r: 3π=V r
4. (a) Write 634 000 000 in Scientific Notation (b) Write 59.12 10−× in Standard Notation (c) Write 0.000 000 55 in Scientific Notation (d) Write 87.25 10× in Standard Notation 5. (a) Calculate 83.2 10× multiplied by 55.3 10−× (b) Calculate 82.8 10× divided by 127.2 10−× (c) Add 87.25 10× and 42.61 10×
Centre for Teaching and Learning | Academic Practice | Academic Skills | Digital Resources Page 1 +61 2 6626 9262 | [email protected] | www.scu.edu.au/teachinglearning [last edited on 7 September 2017]
Topic 1: Exponent rules - whole number exponents The product rule To help develop the rule, let’s consider the question below:
5 4
5 factors 4 factors
altogether gives 9 factors
9
2 2
(2 2 2 2 2) (2 2 2 2)
2 2 2 2 2 2 2 2 22
×
= × × × × × × × ×
= × × × × × × × ×=
Generalising this:
m n m na a a +× =
(When multiplying two powers with the same base, add the exponents.) Examples:
m n m na a a −÷ = (When dividing two powers with the same base, subtract the exponents.)
Examples:
8 6
8
6
8 6
2
5 55555 or 25
−
÷
=
==
( ) ( )( )( )( )( )
9 5
9
5
9 5
4
3 3
3
3
3
3 or 81
− −
−
−
−−
−
÷
=
=
=
15 3
15
3
15 3
12
p ppppp
−
÷
=
==
5 42 3cannot be done
the bases are not the same
÷
10 5
10 5
5
(2 ) (2 )(2 )(2 )
b bbb
−÷
==
4
4
b
bc c
c −÷
=
7 22 2cannot be done
the rule is for quotients
−
8 5
1 8
2 5
8 5
3
4 848
2
2
x xxx
x
x
−
÷
=
=
=
8 3 5 3
312 4
12a b a b÷
=8 3
1 5 3
8 5 3 3
3 0
0
3
43
13
As 1 (see next section)3
a ba b
a b
a bba
− −
=
==
=
Page 3
Zero exponent rule To help develop the rule, let’s consider the question below where the solution is obtained by exponents and by evaluating the exponents:
5 5
5
5
5 5
0
2 22222
−
÷
=
==
or alternatively
5 5
5
5
2 22232321
÷
=
=
=
means that 02 1=
Generalising this for many different examples gives:
0 1a = (A power with a zero exponent is equal to 1.)
Examples:
05 1= ( )03 1− =
5 0
5
51
p ppp
×= ×=
0(5 ) 1a = 05
5 15
a= ×=
( )02 841a c
=
6 3 6
416 4
16x y x y÷
=6 3
1 6 1
6 6 3 1
0 2
2
4444
x yx y
x yx yy
− −===
01 234 567 1= 3 3
1 1 3
1 1 3
1 1 3 3
0 0
4 444
1
ab aba ba b
a ba b
− −
÷
=
===
Page 4
The power rule To help develop the rule, let’s consider the question below:
( )35
5 factors 5 factors 5 factors
altogether gives 3 5=15 factors
15
2
(2 2 2 2 2) (2 2 2 2 2) (2 2 2 2 2)
2 2 2 2 2 2 2 2 2 2 2 2 2 2 22
×= × × × × × × × × × × × × × ×
= × × × × × × × × × × × × × ×=
Generalising this:
( )nm m na a ×=
(When a power is raised to a power, multiply the exponents.) Examples:
2. Simplify the following (leave your answer in power form).
(a) 2 55 5× (b) 4 5m m× (c) 4 3a a a× ×
(d) 4 010 10 10× × (e) 4
2 4r r× (f) 2 45 3c c×
(g) 2 4 3a b a bc× (h) 3 4 2 24 5x y z x y× (i) 2 3 5 3 2 2 05 5e f g e f g×
(j) 9 54 4÷ (k) 8 5x x÷ (l) 5h h÷
(m) 43 3x ÷ (n) 5
2
164
aa
(o) 2 4
2
525
a ba b
(p) 1 12 4a a÷ (q) ( )342 (r) ( )
14 25
(s) ( )54b (t) ( )43t (u) ( )34 32a b
(v) 3
4y
(w) ( )( )
24
42
2
2
y
x (x) ( )52
b
3. Simplify the following.
(a) ( )42 3a a× (b) ( )24
5
32yy
(c) ( )5 4
32
2x x
x
×
(d) ( )003 3b b+ (e) ( )2 2
4
3 39
n nn×
(f) ( ) ( )32 25 mn mn×
(g) 2 5 7 3 4 2m n p m p n p× × (h) 052 )(4 ba− (i) 432 32 abba ×
(j) 32
45
102
yxyx
(k) 4 23
4
2x yy x
×
(l) 0
0
44
−
aa
(m)
2
8
6
3625
=
qp
(n) ( )=39627 cba (o) ( )( ) ( )32
0
2
32
46
223
xyxy
yxxy
÷
Numeracy
Centre for Teaching and Learning | Academic Practice | Academic Skills | Digital Resources Page 1 +61 2 6626 9262 | [email protected] | www.scu.edu.au/teachinglearning [last edited on 22 November 2017]
Topic 2: Exponent rules – integers and fraction exponents Negative exponents To help develop the rule, let’s consider the question below: 4
32 - both the 4 and the 32 are powers of 2.
This question can be solved two ways: by normal fraction methods and using exponents. Because the starting question is the same, the answers must also be the same.
1432
8
3
1812
=
=
2
5
2 5
3
432
2222
−
−
=
==
This means that: 33
1 22
−=
Generalising this:
1 1 or m mm ma a
a a−
−= =
One good way to remember this rule is to say
‘Take the reciprocal and change the sign of the exponent’
Note: There is a convention in mathematics that answers to questions should be expressed with positive exponents only. When solving a question, use the rules in the previous section first (if possible), if an answer still contains a negative exponent then use the rule above to express with positive exponents. Examples:
Fractional exponents To help develop the rule, let’s consider the information below:
We know using exponents that: 1 1 1 1 1 1
13 3 3 3 3 3x x x x x x+ +
× × = = = From the ideas about roots of numbers: 3 3 3x x x x× × =
That means that 1
33x x= Generalising this:
1
nna a= (This is for unit fractions)
Before proceeding with some examples, let’s have a look at the meaning of different roots.
Square root: the square root of a number is usually written as n but more correctly should be written as 2 n , the 2 indicating a relationship to squaring. Square root of a number: for example 36 6= , because 26 6 6 36× = = .
Cube root of a number: for example 3 125 5= , because 35 5 5 5 125× × = = .
Forth root of a number: for example 4 16 2= , because 42 2 2 2 2 16× × × = = .
Fifth root of a number: for example 5 243 3= , because 53 3 3 3 3 3 243× × × × = = . etc.
Using Calculators Calculators vary in the way roots are calculated. Most calculators find square roots by: s25=5 Most calculators find cube roots by: S8=2 To find any root, most calculators use x To find 4 625 , do 4F625=5 Another option is to use the ‘raising to an exponent’ key (^ or w) using the fraction or changing the
fraction to a decimal: ( ) ( )1 0.254 4625 625 625= = , do 625^0.25=5 or
625^a1 4=5 Examples:
Page 4
122525
5==
14
4
256256
4==
( )
14
18 4
184
2
256
2
22 or 4
×
=
==
( )1420
2.115 (to 3 d.p.) By calculator=
( )1
6 3
163
2
p
pp
×==
( )1
52
52
1.5153
1.51531.008 to 3 d.p.
By calculator
==
129
169
169
1634
=
=
=
( )1
4 2
1 142 2
2
2
9
99
3
x
xx
x
×===
64
64
1642
32
6464
88
xx
xx
×
=
==
12
12
1001
1001100
110
−
=
=
=
14 4
8
14 4 4
4 8
1 14 44 4
1 14 84 4
1 1
1 2
2
1681
23
2
32323
xy
xy
x
yxyxy
× ×
× ×
=
=
=
=
14 4
8
144 4
184 4
1
2
2
1681
16
812323
xy
x
yxyxy
×
×
=
=
=
If the exponent is a non-unit fraction, the power of a power rule can assist.
( )2 1
2 233 3a a a= =
Page 5
Generalising this:
( ) or m m
mn nna a a=
Examples:
Evaluate 238
This question can be done in three ways.
In the first method, 8 is written in power form as 32and the power of a power
rule is used.
The second method uses a calculator with a fraction
exponent.
The third method involves changing the fraction exponent to
a decimal.
( )1
1
23
23 3
233
2
8
2
224
×
=
===
( )
23
12 3
23
8
8
8
=
=
S8d=4
23
0.6666666666668
84
==
8^a2 3=4
8^.666666=4
It is important to fill the display of your calculator with the recurring digit. Be careful with calculating with recurring decimals.
Evaluate 35243
( )1
1
35
35 5
355
3
243
3
3327
×
=
===
( )
35
13 5
5 3
243
243
243
=
=
5F243D=27
35
0.6243
24327
==
243^0.6=27
or
243w0.6=27
Evaluate 2550
2550
As 50 cannot be written in power form, only a calculator
solution is possible.
50^a2 5=
25
5 2
5050=
5F50d=
4.781762499
25
0.450
50=
50^0.4=
4.781762499
Page 6
4.781762499
Evaluate 47128
−
( )1
1
47
47
47 7
477
4
1281
1281
21
2121 or 0.0625
16
−
×
=
=
=
=
=
47
7 4
128128
−
−=
7F128^z4=0.0625
47
0.571428571128
1280.0625
−
−==
128^z0.57
1428571=0.0625
or
128^az47=0.0625
Simplify ( )3
4 8 4625a b
( )1 1 2
1 1 1
34 8 4
3 3 34 4 84 4 4
3 3 6
3 6
625
55125
a b
a ba b
a b
× × ×
===
( )1 2
1 1
34 8 4
3 34 834 4 4
3 6
625
625125
a b
a ba b
× ×
==
4F625^3=125
( )1 2
1 1
34 8 4
3 34 80.75 4 4
3 6
625
625125
a b
a ba b
× ×
==
625^0.75=125
Simplify 3 3 564a b
( )1 1
1 1
3 3 5
13 5 3
1 1 13 3 53 3 3
53
64
64
4
4
a b
a b
a b
ab
× × ×
=
=
=
( )
11
3 3 5
13 53 3
1 13 53 3
53
64
64
4
4
a b
a b
a b
ab
× ×
=
=
=
3S64=4
( )1
1
3 3 5
13 5 3
1 13 50.333 3 3
53
64
64
64
4
a b
a b
a b
ab
× ×
=
=
=
64^0.33333
3333=4
Page 7
Simplify
36 4
8
625ab
−
21
1 31 2
38 4
6
38 4
4 6
384
3 34 64 4
6 6
9 93 2 2
625
5
5
or 5 125
ba
ba
b
ab b
a a
×
× ×
=
=
=
2
1
32
38 4
6
348
364
384
3634 4
6
92
625
625
625
125
ba
b
a
b
ab
a
×
×
=
=
=
625^a34=125
4F625^3
=125
21
32
38 4
6
348
364
384
360.75 4
6
92
625
625
625
125
ba
b
a
b
ab
a
×
×
=
=
=
625^0.75=125
Video ‘Exponent Rules – Integer & Fraction Exponents’
Centre for Teaching and Learning | Academic Practice | Academic Skills | Digital Resources Page 1 +61 2 6626 9262 | [email protected] | www.scu.edu.au/teachinglearning [last edited on 7 September 2017]
Topic 3: Equations with exponents Before starting to solve equations containing exponents, it is important to think about opposite operations just like in the Equations Topic. You will recall the opposites below:
Operation Opposite Operation
Addition + Subtraction -
Subtraction - Addition +
Multiplication x Division ÷
Division ÷ Multiplication x
Squaring 2n Square root n
Square root n Squaring 2n
Revision example:
2
2
2
2
2
2
2 3 112 3 3 11 3 take 3 from both sides
2 82 8 divide both sides by 22 2
44 take square root of both sides
2
xx
xx
xxx
+ =+ − = − →
=
= →
=
= ± →= ±
2
2
2
2
2
2
2 3 112 11 3 +3 becomes -3 on the RHS2 8
8 2 becomes 2 on the RHS24
4 becomes on the RHS2
xxx
x
xxx
+ == − →=
= →× ÷
== ± →= ±
There are two solutions here because both 2 and -2 satisfy the equation. This always occurs with even roots.
This list can be expanded to cubes, cube roots, to the 4, forth root, etc.
Operation Opposite Operation
Cubing ( ) 3 Cube root ( 3 ) or 13
To the 4th power Forth root ( 4 ) or 14
To the 5th power Fifth root ( 5 ) or 15
To the 6th power Sixth root ( 6 ) or 16
To the 7th power Seventh root ( 7 ) or 17
etc. etc.
To the power 23
To the power 32
To the power 35
To the power 53
Examples: Solve for x
Both sides method Short -cut Method 4
44 4
256256 take the fourth root of both sides4
xx
x
=
= →= ±
4
4 44
256256 becomes on the RHS4
xxx
== →= ±
( )6
6 66
3
3 raise both sides to the power of 6729
x
xx
=
=
=
6
6 66
33 becomes on the RHS729
xxx
== →=
( )
5
5
5
5
5
555
3 1 53 1 1 5 1 add 1 to both sides
3 63 6 divide both sides by 3
3 32
2 raise both side to the power of 532
xx
xx
x
xx
− =− + = + →
=
= →
=
= →
=
( )
5
5
5
5
5
55 5
3 1 53 5 1 - 1 becomes + 1 on the RHS3 6
6 by 3 becomes by 3 on the RHS 322 becomes on the RHS32
xxx
x
xxx
− == + →=
= →× ÷
== →=
There are two solutions here because both 4 and -4 satisfy the equation. This always occurs with even roots
Page 3
2
1 2
1
4 404 40
4
xx
−
−
=
=
( )
10
1
2
112 22
12
divide both sides by 44
110 raise both sides to the 2
101
101 100.3162 ( to 4 d.p.)
x
x
x
x
x
−
−−−
→
= → −
=
=
=
= ±
2
2
4 4040
x
x
−
−
=
=10
1
2
1 122 2
4 becomes 4 on the RHS4
10
10 becomes on the RHS1100.3162 ( to 4 d.p.)
x
x
x
x
−
− −−
→ × ÷
=
= →
=
= ±
( )( )
2
2
2 1 49
2 1 49 take square root of both sides2 1 7
2 1 1 7 1 or 7 1 add 1 to both sides2 8 or 62 8 6 or divide both sides by 22 2 2
4 or 3
x
xx
xxx
x
− =
− = →− = ±
− + = + − + →= −
−= →
= −
( )2
2
2 1 492 1 49 becomes on the RHS2 1 7
2 7 1 or 7 1 1 becomes 1 on the RHS2 8 or 6
8 6 or 2 becomes 2 on the RHS2 24 or 3
xxx
xx
x
x
− =− = →− = ±
= + − + → − += −
−= → × ÷
= −
In the question below, the opposite of raising a number to a power is raising it to the reciprocal power. Remember a number multiplied by its reciprocal gives 1.
In the question below, the fraction exponent becomes the reciprocal on the other side of the equation.
23
32 323 2
3
50
50
50353.55
x
x
xx
=
=
==
23
32
3
50
5050
353.55
x
xxx
=
=
==
It is also important to be able to rearrange formulae that contain powers.
Rearrange the formulae below to make the pronumeral in brackets the subject. [ ]2
1 2
1
2
2
divide both sides by
take the square root of both sides
A r rA r
A r
A r
Ar
ππ π
π π
π
π
π
=
= →
= →
=
=
[ ]2
2
2
becomes on LHS
becomes on the LHS
A r rA r
A r
Ar
π
π ππ
π
π
=
= →× ÷
= →
=
Page 4
[ ]3
3 11
3
4 The first step is to remove the fraction343 3 Multiple both sides by 33
3 43 44
V r r
V r
V rV
π
π
ππ
π
=
= × →
=
=3
4rπ
3
3 33
3
Divide both sides by 4
343 take cube root of both sides4
34
V r
V r
Vr
π
π
π
π
→
=
= →
=
[ ]3
3
3
3 33
3
4 The first step is to remove the fraction3
3 4 3 becomes 3 on LHS3 4 becomes 4 on LHS43 becomes on the LHS4
34
V r r
V rV r
V r
Vr
π
π
π ππ
π
π
=
= → ÷ ×
= → × ÷
= →
=
[ ]1 2
1
2
f CLC
f LCLC
π
π
=
× = LC×
( )2
2
2 2
2 2
2 2
multiply both sides by to get C into the numerator
2. Rearrange these formulae for the pronumeral in brackets
(a) [ ]24 A r rπ= (b) [ ]2 V r h rπ= (c) [ ]2 2 2 c a b b= +
(d) [ ]21 2
F mv v= (e) [ ]2 Q SLd d= (f) [ ]2 2 2 v u as u= +
(g) [ ]2 2 2 Z R L Lω= + (h) [ ]2
2 GMR cc
= (i) [ ]2
2 EIP LL
π=
The next questions are challenging, take care!
(j) [ ]3
2 rT rGM
π= (k) [ ]2
vF m g vR
= −
(l) ( ) [ ]
42 1
8r p p
R rnL
π −=
(m) [ ](1 ) nA P i i= +
Numeracy
Centre for Teaching and Learning | Academic Practice | Academic Skills | Digital Resources Page 1 +61 2 6626 9262 | [email protected] | www.scu.edu.au/teachinglearning [last edited on 7 September 2017]
Topic 4: Scientific notation Scientific Notation is a way of representing numbers that are very large or very small usually to avoid writing down numbers that contain many zeros. When numbers are written in scientific notation they usually written in this form:
10en × The number 4 000 000 can be written as 4 1000 000× and then 64 10× .
The number 0.000 8 can be written as 8
10 000 and then 4
48 8 10
10−= × .
The number 532 000 can be written as 55.32 100 000 5.32 10× = × . Mostly though, students look at the number of places that the decimal point has moved. Writing numbers in Scientific Notation Write 395 000 in scientific notation.
Step 1 From your number, obtain the number between 1 and 9.9999999 ?395 000 3.95 10= ×
Step 2 Calculate how many places the decimal point moved.
?395 000 3.95 10= × The decimal point moved 5 places, so the ? becomes 5.
5395 000 3.95 10= ×
Step 3 The exponent can be positive, negative or 0. In this step, the sign of the exponent is checked. If the original number is smaller than 1, then the exponent is negative. Otherwise it is positive or possibly zero.
Centre for Teaching and Learning | Academic Practice | Academic Skills | Digital Resources Page 1 +61 2 6626 9262 | [email protected] | www.scu.edu.au/teachinglearning [last edited on 22 November 2017]
Topic 5: Operations with numbers in scientific notation Multiplying When multiplying numbers in scientific notation, the number part is multiplied and the exponent part is multiplied using exponent rules. Example
( ) ( )( ) ( )
6 2
6 2
6 2
4
3.1 10 1.4 103.1 1.4 10 10 rearrange
4.34 104.34 10
−
−
−
+
× × ×
= × × × →
= ×= ×
It is important to look at the solution to see if the answer is still in scientific notation. In the example below, an adjustment is required to put the answer into scientific notation.
( ) ( )( ) ( )
5 9
5 9
5 9
14
1 14 1
1 14
13
5.2 10 3.5 105.2 3.5 10 10 rearrange
18.2 1018.2 10 The number 18.2 is not in the range 1 to 9.9999 , 1.82 10 10 18.2 1.82 10 1.82 101.82 10
−
−
− −
− −
− +
−
−
+
−
× × ×
= × × × →
= ×= × → …= × × → = ×= ×= ×
The speed of light is approximately 83 10× m/sec. How far will it travel in 5 hours?
Dividing Dividing numbers in scientific notation is similar to multiplication. The number part is divided as usual and the exponent part is divided using exponent rules. Example:
7
2
7 2
7 2
9
6.25 105.1 10
(6.25 5.1) (10 10 )1.2255 101.2255 10
−
−
−
−
××
= ÷ × ÷= ×= ×
As in multiplying, it is important to look at the solution to see if the answer is still in scientific notation. In the example below, an adjustment is required to put the answer into scientific notation.
7
4
7 4
7 4
11
1 11 1
12
3.12 107.36 10
(3.12 7.36) (10 10 )0.4239 100.4239 10 The number 0.4239 is not in the range 1 to 9.9999 ,4.239 10 10 0.4239=4.239 104.239 10
−
−
− −
−
− − −
−
××
= ÷ × ÷= ×= × → …= × × → ×= ×
The mass of a hydrogen atom is 241.6 10−× grams. How many
atoms are there in 1mg ( 310− gram) of hydrogen? 3
Adding and Subtracting When adding or subtracting any numbers, it is important that numbers of the same place value are added or subtracted. When adding or subtracting whole numbers or decimals these means lining up the numbers in place value rows. With numbers written in scientific notation, lining up numbers in place values is achieved by having exponent parts the same. Add 5 7(3 10 ) and (4 10 )× ×
5 7
2 7 7
7 7
7
(3 10 ) + (4 10 )3 10 10 4 100.03 10 4 10 Both numbers are in the same place value4.03 10
One exponent must be changed to be the same as the other exponent. Choosing the higher exponent makes the finding then solution slightly easier.
Remember 4− is larger than 7− .
Page 4
Calculator use To use scientific notation on your calculator, some calculators have a key marked Exp , others K.
112 10× is entered by the sequence: 2K11 or 2 Exp 11
64.25 10−× is entered by the sequence 4.25Kz6 or 4.25 Exp z6
158.75 10− × is entered by the sequence z8.75K15
191.6 10−− × is entered by the sequence z1.6Kz19 If you enter a number expressed in scientific notation using Exp or the K keys, the calculator treats this as a single number (which it is) . However, if you enter your number using the G key, for example 2OG11, the calculator treats this as two numbers being multiplied and this can lead to incorrect answers when performing calculations . Using your calculator to change between scientific notation and standard notation can be quite difficult. If you require this you will need to obtain the instructions for your calculator and find out how to achieve this. Performing an operation such as 11 62 10 4.25 10−× × × is entered as:
2K11[4.25Kz6=850000 The format of your answer will vary from calculator to calculator. Because the answer to this question is able to be displayed on a calculator in standard notation, the answer displayed could be 850 000, so you have to change this to 58.5 10× if you specifically need the answer in scientific notation. Other calculators may automatically give the answer in scientific notation. If numbers are outside the range of what can be displayed in standard form on the calculator, your calculator should display the answer in scientific notation. Even when using a calculator, it is still important to be able to change from standard form to scientific notation and vice-versa.
Centre for Teaching and Learning | Academic Practice | Academic Skills | Digital Resources Page 1 +61 2 6626 9262 | [email protected] | www.scu.edu.au/teachinglearning [last edited on 7 September 2017]
Answers to activity questions Check your skills 1.
(a)
6 3
6 3 1
10
444
x x xxx
+ +× ×
==
(b)
6 0
6
61
a aaa
÷= ÷=
(c)
( ) ( )2 43 4 2
6 4
4 8
1 10
4 24
2416
x x xx x
xx
× ÷
×=
= 4 8
10 8
24
4
xx
x
−
=
=
2.
(a)
( )41
22 02
1 42 2 2
4 2
22
1
1 or
x x x
x xx x
xx
−
×− ×
−
−
× ×
= × ×= ×
=
(b)
( )
3
19 3
193
3
512
2
228
×
=
===
(c)
( )( )
29 6 3
23 9 6 3
2 2 23 9 63 3 3
6 4
27
3
39
x y
x y
x yx y
× × ×
=
==
3. (a)
4
4
8181
3
xxx
== ±= ±
(b)
4
4
4
4
4
4
3 1 73 7 13 6
632216
xxx
x
xxx
+ == −=
=
===
(c)
3
3
3
V rV r
V r
π
π
π
=
=
=
4. (a) 634 000 000 in Scientific Notation is 86.34 10× (b) 59.12 10−× in Standard Notation is 0.000 091 2 (c) 0.000 000 55 in Scientific Notation is 75.5 10−× (d) 87.25 10× in Standard Notation is 725 000 000