Transcript N2 Powers, roots and standard form

KS4 Mathematics
N2 Powers, roots and
standard form
1 of 70
© Boardworks Ltd 2005
Contents
N2 Powers, roots and standard form
A N2.1 Powers and roots
A N2.2 Index laws
A N2.3 Negative indices and reciprocals
A N2.4 Fractional indices
A N2.5 Surds
A N2.6 Standard form
2 of 70
© Boardworks Ltd 2005
Square numbers
When we multiply a number by itself we say that we are
squaring the number.
To square a number we can write a small 2 after it.
For example, the number 3 multiplied by itself can be
written as
3×3
or
32
The value of three squared is 9.
The result of any whole number multiplied by itself is called
a square number.
3 of 70
© Boardworks Ltd 2005
Square roots
Finding the square root is the inverse of finding the square:
squared
8
64
square rooted
We write
64 = 8
The square root of 64 is 8.
4 of 70
© Boardworks Ltd 2005
The product of two square numbers
The product of two square numbers is always another
square number.
For example,
4 × 25 = 100
because
2×2×5×5=2×5×2×5
and
(2 × 5)2 = 102
We can use this fact to help us find the square roots of larger
square numbers.
5 of 70
© Boardworks Ltd 2005
Using factors to find square roots
If a number has factors that are square numbers then we
can use these factors to find the square root.
For example,
Find 400
6 of 70
Find 225
√400 = √(4 × 100)
√225 = √(9 × 25)
= √4 × √100
= √9 × √25
= 2 × 10
=3×5
= 20
= 15
© Boardworks Ltd 2005
Finding square roots of decimals
We can also find the square root of a number can be made
be dividing two square numbers.
For example,
7 of 70
Find 0.09
Find 0.0144
0.09 = (9 ÷ 100)
0.0144 = (144 ÷ 10000)
= √9 ÷ √100
= √144 ÷ √10000
= 3 ÷ 10
= 12 ÷ 100
= 0.3
= 0.12
© Boardworks Ltd 2005
Approximate square roots
If a number cannot be written as a product or quotient of two
square numbers then its square root cannot be found exactly.
Use the 
key on your calculator to find out 2.
The calculator shows this as 1.414213562
This is an approximation to 9 decimal places.
The number of digits after the decimal point is infinite and
non-repeating.
This is an example of an irrational number.
8 of 70
© Boardworks Ltd 2005
Estimating square roots
What is 50?
50 is not a square number but lies between 49 and 64.
Therefore,
49 < 50 < 64
So,
7 < 50 < 8
Use the 
50 is much
closer to 49 than
to 64, so 50 will
be about 7.1
key on you calculator to work out the answer.
50 = 7.07 (to 2 decimal places.)
9 of 70
© Boardworks Ltd 2005
Negative square roots
5 × 5 = 25
and
–5 × –5 = 25
Therefore, the square root of 25 is 5 or –5.
When we use the  symbol we usually mean the positive
square root.
We can also write ± to mean both the positive and the
negative square root.
However the equation,
x2 = 25
has 2 solutions,
x=5
10 of 70
or
x = –5
© Boardworks Ltd 2005
Squares and square roots from a graph
11 of 70
© Boardworks Ltd 2005
Cubes
The numbers 1, 8, 27, 64, and 125 are all:
Cube numbers
13 = 1 × 1 × 1 = 1
‘1 cubed’ or ‘1 to the power of 3’
23 = 2 × 2 × 2 = 8
‘2 cubed’ or ‘2 to the power of 3’
33 = 3 × 3 × 3 = 27
‘3 cubed’ or ‘3 to the power of 3’
43 = 4 × 4 × 4 = 64
‘4 cubed’ or ‘4 to the power of 3’
53 = 5 × 5 × 5 = 125
‘5 cubed’ or ‘5 to the power of 3’
12 of 70
© Boardworks Ltd 2005
Cube roots
Finding the cube root is the inverse of finding the cube:
cubed
5
125
cube rooted
We write
125 = 5
3
The cube root of 125 is 5.
13 of 70
© Boardworks Ltd 2005
Squares, cubes and roots
14 of 70
© Boardworks Ltd 2005
Index notation
We use index notation to show repeated multiplication by the
same number.
For example:
we can use index notation to write 2 × 2 × 2 × 2 × 2 as
Index or power
25
base
This number is read as ‘two to the power of five’.
25 = 2 × 2 × 2 × 2 × 2 = 32
15 of 70
© Boardworks Ltd 2005
Index notation
Evaluate the following:
62 = 6 × 6 = 36
34 = 3 × 3 × 3 × 3 = 81
(–5)3 = –5 × –5 × –5 = –125
When we raise a
negative number to
an odd power the
answer is negative.
27 = 2 × 2 × 2 × 2 × 2 × 2 × 2 = 128
(–1)5 = –1 × –1 × –1 × –1 × –1 = –1
(–4)4 = –4 × –4 × –4 × –4 = 256
16 of 70
When we raise a
negative number to
an even power the
answer is positive.
© Boardworks Ltd 2005
Using a calculator to find powers
We can use the xy key on a calculator to find powers.
For example:
to calculate the value of 74 we key in:
7
xy
4
=
The calculator shows this as 2401.
74 = 7 × 7 × 7 × 7 = 2401
17 of 70
© Boardworks Ltd 2005
Contents
N2 Powers, roots and standard form
A N2.1 Powers and roots
A N2.2 Index laws
A N2.3 Negative indices and reciprocals
A N2.4 Fractional indices
A N2.5 Surds
A N2.6 Standard form
18 of 70
© Boardworks Ltd 2005
Multiplying numbers in index form
When we multiply two numbers written in index form and with
the same base we can see an interesting result.
For example:
34 × 32 = (3 × 3 × 3 × 3) × (3 × 3)
=3×3×3×3×3×3
= 36 = 3(4 + 2)
73 × 75 = (7 × 7 × 7) × (7 × 7 × 7 × 7 × 7)
=7×7×7×7×7×7×7×7
= 78 = 7(3 + 5)
When we multiply two numbers with the same base the
What do you notice?
indices are added. In general, xm × xn = x(m + n)
19 of 70
© Boardworks Ltd 2005
Dividing numbers in index form
When we divide two numbers written in index form and with
the same base we can see another interesting result.
For example:
45
56
÷
42
4×4×4×4×4
=
= 4 × 4 × 4 = 43 = 4(5 – 2)
4×4
÷
54
5×5×5×5×5×5
=
= 5 × 5 = 52 = 5(6 – 4)
5×5×5×5
When we divide two numbers with the same base the
What do you notice?
indices are subtracted. In general, xm ÷ xn = x(m – n)
20 of 70
© Boardworks Ltd 2005
Raising a power to a power
Sometimes numbers can be raised to a power and the result
raised to another power.
For example,
(43)2 = 43 × 43
= (4 × 4 × 4) × (4 × 4 × 4)
= 46 = 4(3 × 2)
When a number is raised to a power and then raised to another
What do you notice?
power, the powers are multiplied. In general, (xm)n = xmn
21 of 70
© Boardworks Ltd 2005
Using index laws
22 of 70
© Boardworks Ltd 2005
The power of 1
Find the value of the following using your calculator:
61
471
0.91
–51
01
Any number raised to the power of 1 is equal to the
number itself. In general, x1 = x
Because of this we don’t usually write the power when a
number is raised to the power of 1.
23 of 70
© Boardworks Ltd 2005
The power of 0
Look at the following division:
64 ÷ 64 = 1
Using the second index law,
64 ÷ 64 = 6(4 – 4) = 60
That means that:
60 = 1
Any non-zero number raised to the power of 0 is equal to 1.
For example,
100 = 1
24 of 70
3.4520 = 1
723 538 5920 = 1
© Boardworks Ltd 2005
Index laws
Here is a summery of the index laws you have met so far:
xm × xn = x(m + n)
xm ÷ xn = x(m – n)
(xm)n = xmn
x1 = x
x0 = 1 (for x = 0)
25 of 70
© Boardworks Ltd 2005
Contents
N2 Powers, roots and standard form
A N2.1 Powers and roots
A N2.2 Index laws
A N2.3 Negative indices and reciprocals
A N2.4 Fractional indices
A N2.5 Surds
A N2.6 Standard form
26 of 70
© Boardworks Ltd 2005
Negative indices
Look at the following division:
3×3
3×3×3×3
32 ÷ 34 =
=
1
1
= 2
3×3
3
Using the second index law,
32 ÷ 34 = 3(2 – 4) = 3–2
That means that
Similarly,
27 of 70
6–1 =
1
6
3–2 =
1
32
7–4 =
1
74
and
5–3 =
1
53
© Boardworks Ltd 2005
Reciprocals
A number raised to the power of –1 gives us the reciprocal
of that number.
The reciprocal of a number is what we multiply the number
by to get 1.
1
The reciprocal of a is
a
a
b
The reciprocal of
is
b
a
We can find reciprocals on a calculator using the x-1 key.
28 of 70
© Boardworks Ltd 2005
Finding the reciprocals
Find the reciprocals of the following:
1) 6
1
The reciprocal of 6 =
6
3
2)
7
3
7
The reciprocal of
=
7
3
3) 0.8
4
0.8 =
5
4
5
The reciprocal of
=
5
4
or
0.8–1 = 1.25
29 of 70
or
6-1
or
3
7
1
=
6
–1
7
=
3
= 1.25
© Boardworks Ltd 2005
Match the reciprocal pairs
30 of 70
© Boardworks Ltd 2005
Index laws for negative indices
Here is a summery of the index laws for negative indices.
x–1 = 1
x
x–n
31 of 70
= 1n
x
The reciprocal of x is
The reciprocal of
xn
1
x
1
is n
x
© Boardworks Ltd 2005
Contents
N2 Powers, roots and standard form
A N2.1 Powers and roots
A N2.2 Index laws
A N2.3 Negative indices and reciprocals
A N2.4 Fractional indices
A N2.5 Surds
A N2.6 Standard form
32 of 70
© Boardworks Ltd 2005
Fractional indices
1
2
Indices can also be fractional. Suppose we have 9 .
1
2
1
2
9 ×9 = 9
In general,
33 of 70
1
2
= 91 = 9
Because 3 × 3 = 9
x 2 = x
1
In general,
But,
+
9 × 9 = 9
But,
Similarly,
1
2
1
3
1
3
1
3
8 ×8 ×8 = 8
1
1
3+ 3
+
1
3
= 81 = 8
8 × 8 × 8 = 8
3
3
3
Because
2×2×2=8
x 3 = x
1
3
© Boardworks Ltd 2005
Fractional indices
3
2
What is the value of 25 ?
3
2
We can think of 25 as 25
1
2
×3
.
Using the rule that (xa)b = xab we can write
25
1
2
×3
= (25)3
= (5)3
= 125
In general,
m
n
n
x = (x)m
34 of 70
© Boardworks Ltd 2005
Evaluate the following
49 12 = √49 = 7
1) 49 12
2) 1000
1000 23 = (3√1000)2 = 102 = 100
3)
1
8 3
1
83
4)
2
64- 3
64-
5) 4
35 of 70
2
3
5
2
1
1
1
= 1 = 3
=
√8
83
2
2
3
1
1
1
1
=
= 2 =
2 =
3
2
3
( √64)
64
4
16
4 52 = (√4)5 = 25 = 32
© Boardworks Ltd 2005
Index laws for fractional indices
Here is a summery of the index laws for fractional indices.
x = x
1
2
n
x = x
1
n
m
n
x =
36 of 70
n
xm
n
or (x)m
© Boardworks Ltd 2005
Contents
N2 Powers, roots and standard form
A N2.1 Powers and roots
A N2.2 Index laws
A N2.3 Negative indices and reciprocals
A N2.4 Fractional indices
A N2.5 Surds
A N2.6 Standard form
37 of 70
© Boardworks Ltd 2005
Surds
The square roots of many numbers cannot be found exactly.
For example, the value of √3 cannot be written exactly as a
fraction or a decimal.
The value of √3 is an irrational number.
For this reason it is often better to leave the square root sign
in and write the number as √3.
√3 is an example of a surd.
Which one of the following is not a surd?
√2, √6 , √9 or √14
9 is not a surd because it can be written exactly.
38 of 70
© Boardworks Ltd 2005
Multiplying surds
What is the value of √3 × √3?
We can think of this as squaring the square root of three.
Squaring and square rooting are inverse operations so,
√3 × √3 = 3
In general, √a × √a = a
What is the value of √3 × √3 × √3?
Using the above result,
√3 × √3 × √3 = 3 × √3
Like algebra, we do
not use the × sign
when writing surds.
= 3√3
39 of 70
© Boardworks Ltd 2005
Multiplying surds
Use a calculator to find the value of √2 × √8.
What do you notice?
√2 × √8 = 4
(= √16)
4 is the square root of 16 and 2 × 8 = 16.
Use a calculator to find the value of √3 × √12.
√3 × √12 = 6 (= √36)
6 is the square root of 36 and 3 × 12 = 36.
In general, √a × √b = √ab
40 of 70
© Boardworks Ltd 2005
Dividing surds
Use a calculator to find the value of √20 ÷ √5.
What do you notice?
√20 ÷ √5 = 2 (= √4)
2 is the square root of 4 and 20 ÷ 5 = 4.
Use a calculator to find the value of √18 ÷ √2.
√18 ÷ √2 = 3 (= √9)
3 is the square root of 9 and 18 ÷ 2 = 9.
In general, √a ÷ √b =
41 of 70

a
b
© Boardworks Ltd 2005
Simplifying surds
We are often required to simplify surds by writing them in the
form a√b. For example,
Simplify √50 by writing it in the form a√b.
Start by finding the largest square number that divides into 50.
This is 25. We can use this to write:
√50 = √(25 × 2)
= √25 × √2
= 5√2
42 of 70
© Boardworks Ltd 2005
Simplifying surds
Simplify the following surds by writing
them in the form a√b.
1) √45
2) √24
3) √300
√45 = √(9 × 5)
√24 = √(4 × 6)
√300 = √(100 × 3)
= √9 × √5
= √4 × √6
= √100 × √3
= 3√5
= 2√6
= 10√3
43 of 70
© Boardworks Ltd 2005
Simplifying surds
44 of 70
© Boardworks Ltd 2005
Adding and subtracting surds
Surds can be added or subtracted if the number under the
square root sign is the same. For example,
Simplify √27 + √75.
Start by writing √27 and √75 in their simplest forms.
√27 = √(9 × 3)
√75 = √(25 × 3)
= √9 × √3
= √25 × √3
= 3√3
= 5√3
√27 + √75 = 3√3 + 5√3 = 8√3
45 of 70
© Boardworks Ltd 2005
Perimeter and area problem
The following rectangle has been drawn on a square grid.
Use Pythagoras’ theorem to find the length and width of the
rectangle and hence find its perimeter and area in surd form.
1
6
2√10
3
√10
2
Width = √(32 + 12)
= √(9 + 1)
= √10 units
Length = √(62 + 22)
= √(36 + 4)
= √40
= 2√10 units
46 of 70
© Boardworks Ltd 2005
Perimeter and area problem
The following rectangle has been drawn on a square grid.
Use Pythagoras’ theorem to find the length and width of the
rectangle and hence find its perimeter and area in surd form.
1
2√10
3
√10
Perimeter = √10 + 2√10 +
6
2
√10 + 2√10
= 6√10 units
Area = √10 × 2√10
= 2 × √10 × √10
= 2 × 10
= 20 units2
47 of 70
© Boardworks Ltd 2005
Rationalizing the denominator
When a fraction has a surd as a denominator we usually
rewrite it so that the denominator is a rational number.
This is called rationalizing the denominator.
5
Simplify the fraction
√2
Remember, if we multiply the numerator and the denominator
of a fraction by the same number the value of the fraction
remains unchanged.
In this example, we can multiply the numerator and the
denominator by √2 to make the denominator into a whole
number.
48 of 70
© Boardworks Ltd 2005
Rationalizing the denominator
When a fraction has a surd as a denominator we usually
rewrite it so that the denominator is a rational number.
This is called rationalizing the denominator.
5
Simplify the fraction
√2
×√2
5
5√2
=
2
√2
×√2
49 of 70
© Boardworks Ltd 2005
Rationalizing the denominator
Simplify the following fractions by
rationalizing their denominators.
2
1)
√3
×√3
2
2√3
=
3
√3
×√3
50 of 70
√2
2)
√5
3
3)
4√7
×√5
√2
√10
=
5
√5
×√5
×√7
3
3√7
=
28
4√7
×√7
© Boardworks Ltd 2005
Contents
N2 Powers, roots and standard form
A N2.1 Powers and roots
A N2.2 Index laws
A N2.3 Negative indices and reciprocals
A N2.4 Fractional indices
A N2.5 Surds
A N2.6 Standard form
51 of 70
© Boardworks Ltd 2005
Powers of ten
Our decimal number system is based on powers of ten.
We can write powers of ten using index notation.
10 = 101
100 = 10 × 10 = 102
1000 = 10 × 10 × 10 = 103
10 000 = 10 × 10 × 10 × 10 = 104
100 000 = 10 × 10 × 10 × 10 × 10 = 105
1 000 000 = 10 × 10 × 10 × 10 × 10 × 10 = 106 …
52 of 70
© Boardworks Ltd 2005
Negative powers of ten
Any number raised to the power of 0 is 1, so
1 = 100
Decimals can be written using negative powers of ten
0.1 =
1
10
1
101
=
1
0.01 = 100 =
1
102
1
0.001 = 1000 =
0.0001 =
1
10000
0.00001 =
= 10-2
1
103
=
1
100000
0.000001 =
53 of 70
=10-1
= 10-3
1
104
=
1
1000000
= 10-4
1
105
=
= 10-5
1
106
= 10-6 …
© Boardworks Ltd 2005
Very large numbers
Use you calculator to work out the answer to
40 000 000 × 50 000 000.
Your calculator may display the answer as:
2 ×1015
,
2
E
15
or
2 15
What does the 15 mean?
The 15 means that the answer is 2 followed by 15 zeros or:
2 × 1015
54 of 70
= 2 000 000 000 000 000
© Boardworks Ltd 2005
Very small numbers
Use you calculator to work out the answer to
0.0002 ÷ 30 000 000.
Your calculator may display the answer as:
1.5 ×10–12 ,
1.5 E –12 or
1.5
–12
What does the –12 mean?
The –12 means that the 15 is divided by 1 followed by 12 zeros.
1.5 × 10-12 = 0.000000000002
55 of 70
© Boardworks Ltd 2005
Standard form
2 × 1015 and 1.5 × 10-12 are examples of a number written in
standard form.
Numbers written in standard form have two parts:
A number
between 1
and 10
×
A power of
10
This way of writing a number is also called standard index
form or scientific notation.
Any number can be written using standard form, however it is
usually used to write very large or very small numbers.
56 of 70
© Boardworks Ltd 2005
Standard form – writing large numbers
For example, the mass of the planet earth is about
5 970 000 000 000 000 000 000 000 kg.
We can write this in
standard form as a number
between 1 and 10
multiplied by a power of 10.
5.97 × 1024 kg
A number
between 1 and 10
57 of 70
A power of ten
© Boardworks Ltd 2005
Standard form – writing large numbers
How can we write these numbers in standard form?
80 000 000 =
8 × 107
230 000 000 =
2.3 × 108
7.24 × 105
724 000 =
6 003 000 000 =
371.45 =
58 of 70
6.003 × 109
3.7145 × 102
© Boardworks Ltd 2005
Standard form – writing large numbers
These numbers are written in standard form.
How can they be written as ordinary numbers?
5 × 1010 =
50 000 000 000
7.1 × 106 =
7 100 000
4.208 × 1011 =
420 800 000 000
2.168 × 107 =
21 680 000
6.7645 × 103 =
59 of 70
6764.5
© Boardworks Ltd 2005
Standard form – writing small numbers
We can write very small numbers using negative powers of
ten.
For example, the width of this shelled amoeba is 0.00013 m.
We write this in standard
form as:
1.3 × 10-4 m.
A number
between 1 and 10
60 of 70
A negative
power of 10
© Boardworks Ltd 2005
Standard form – writing small numbers
How can we write these numbers in standard form?
0.0006 =
0.00000072 =
0.0000502 =
0.0000000329 =
0.001008 =
61 of 70
6 × 10-4
7.2 × 10-7
5.02 × 10-5
3.29 × 10-8
1.008 × 10-3
© Boardworks Ltd 2005
Standard form – writing small numbers
These numbers are written in standard form.
How can they be written as ordinary numbers?
8 × 10-4 =
0.0008
2.6 × 10-6 =
0.0000026
9.108 × 10-8 =
0.00000009108
7.329 × 10-5 =
0.00007329
8.4542 × 10-2 =
62 of 70
0.084542
© Boardworks Ltd 2005
Which number is incorrect?
63 of 70
© Boardworks Ltd 2005
Ordering numbers in standard form
Write these numbers in order from smallest to largest:
5.3 × 10-4, 6.8 × 10-5, 4.7 × 10-3, 1.5 × 10-4.
To order numbers that are written in standard form start by
comparing the powers of 10.
Remember, 10-5 is smaller than 10-4. That means that 6.8 × 105 is the smallest number in the list.
When two or more numbers have the same power of ten we
can compare the number parts. 5.3 × 10-4 is larger than
1.5 × 10-4 so the correct order is:
6.8 × 10-5,
64 of 70
1.5 × 10-4,
5.3 × 10-4,
4.7 × 10-3
© Boardworks Ltd 2005
Ordering planet sizes
65 of 70
© Boardworks Ltd 2005
Calculations involving standard form
What is 2 × 105 multiplied by 7.2 × 103 ?
To multiply these numbers together we can multiply the
number parts together and then the powers of ten together.
2 × 105 × 7.2 × 103 = (2 × 7.2) × (105 × 103)
= 14.4 × 108
This answer is not in standard form and must be converted!
14.4 × 108 = 1.44 × 10 × 108
= 1.44 × 109
66 of 70
© Boardworks Ltd 2005
Calculations involving standard form
What is 1.2 × 10-6 divided by 4.8 × 107 ?
To divide these numbers we can divide the number parts and
then divide the powers of ten.
(1.2 × 10-6) ÷ (4.8 × 107) = (1.2 ÷ 4.8) × (10-6 ÷ 107)
= 0.25 × 10-13
This answer is not in standard form and must be converted.
0.25 × 10-13 = 2.5 × 10-1 × 10-13
= 2.5 × 10-14
67 of 70
© Boardworks Ltd 2005
Travelling to Mars
How long would it take a space ship travelling at an average
speed of 2.6 × 103 km/h to reach Mars 8.32 × 107 km away?
68 of 70
© Boardworks Ltd 2005
Calculations involving standard form
How long would it take a space ship travelling at an average
speed of 2.6 × 103 km/h to reach Mars 8.32 × 107 km away?
Rearrange
speed =
distance
time
to give
time =
distance
speed
8.32 × 107
Time to reach Mars =
2.6 × 103
= 3.2 × 104 hours
This is
8.32 ÷ 2.6
69 of 70
This is
107 ÷ 103
© Boardworks Ltd 2005
Calculations involving standard form
Use your calculator to work out how long
3.2 × 104 hours is in years.
You can enter 3.2 × 104 into your calculator using the EXP key:
3
.
2
EXP
4
Divide by 24 to give the equivalent number of days.
Divide by 365 to give the equivalent number of years.
3.2 × 104 hours is over 3½ years.
70 of 70
© Boardworks Ltd 2005