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Transcript
```AS-Level Maths:
Core 1
for Edexcel
C1.1 Algebra and
functions 1
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1 of 38
Using and manipulating surds
Using and manipulating surds
Contents
Rationalizing the denominator
The index laws
Zero and negative indices
Fractional indices
Solving equations involving indices
Examination-style questions
2 of 38
Types of number
We can classify numbers into the following sets:
The set of natural numbers, :
Positive whole numbers {0, 1, 2, 3, 4 …}
The set of integers, :
Positive and negative whole numbers {0, ±1, ±2, ±3 …}
The set of rational numbers, :
Numbers that can be expressed in the form mn , where n and
m are integers. All fractions and all terminating and recurring
decimals are rational numbers; for example, ¾, –0.63, 0.2.
The set of real numbers, :
All numbers including irrational numbers; that is, numbers
that cannot be expressed in the form mn , where n and m are
integers. For example,  and 2 .
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Introducing surds
2, √3
3 or
5,
When the square root of a number, for example √2,
or √5,is
irrational, it is often preferable to write it with the root sign.
Numbers written in this form are called surds.
Can you explain why √1.69
1.69 is not a surd?
1.69 is not a surd because it is not irrational.
√1.69
169
1.69 =
100
169
=
100
13
=
10
= 1.3
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This uses the fact that
a
a
=
b
b
Manipulating surds
When working with surds it is important to remember the
following two rules:
ab = a × b
and
a
a
=
b
b
Also:
a× a =a
You should also remember that, by definition, √aa means the
positive square root of a.
5 of 38
Simplifying surds
We are often required to simplify surds by writing them in the
form a b .
We can do this using the fact that
For example:
ab = a × b .
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
6 of 38
Simplifying surds
Simplify the following surds by writing
b.
them in the form aa√b.
45 = 9×5
7 of 38
3) 3 40
2) 98
1) 45
98 = 49× 2
3
40 = 3 8 × 5
= 9× 5
= 49 × 2
= 3 8×3 5
=3 5
=7 2
= 23 5
Simplifying surds
8 of 38
Surds can be added or subtracted if the number under the
square root sign is the same. For example:
Simplify 45 + 80.
Start by writing 45 and 80 in their simplest forms.
45 = 9×5
80 = 16×5
= 9× 5
= 16 × 5
=3 5
=4 5
45 + 80 = 3 5 + 4 5 = 7 5
9 of 38
Expanding brackets containing surds
Simplify the following:
1) (4  2)(1+ 3 2)
2) ( 7  2)( 7 + 2)
= 4 +12 2  2  6
= 7+ 2 7  2 7  2
= 11 2  2
=5
Problem 2) demonstrates the fact that (a – b)(a + b) = a2 – b2.
In general:
( a  b )( a + b )  a  b
10 of 38
Rationalizing the denominator
Contents
Using and manipulating surds
Rationalizing the denominator
The index laws
Zero and negative indices
Fractional indices
Solving equations involving indices
Examination-style questions
11 of 38
Rationalizing the denominator
When a fraction contains a surd as the denominator we
usually rewrite it so that the denominator is a rational number.
This is called rationalizing the denominator. For example:
5
Simplify the fraction
.
2
In this example we rationalize the denominator by multiplying
the numerator and the denominator by 2.
× 2
5
5 2
=
2
2
× 2
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Rationalizing the denominator
Simplify the following fractions by
rationalizing their denominators.
2
1)
3
2)
× 3
2
2 3
=
3
3
× 3
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2
5
3
3)
4 7
× 5
2
=
5
× 5
× 7
10
5
3
3 7
=
28
4 7
× 7
Rationalizing the denominator
When the denominator involves sums of differences between
surds we can use the fact that
(a – b)(a + b) = a2 – b2
to rationalize the denominator. For example:
Simplify
1
5 2
=
1
.
5 2
1
5 2
×
5 +2
5 +2
5 +2
=
54
= 5 +2
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Rationalizing the denominator
More difficult examples may include surds in both the
numerator and the denominator. For example:
2 3 1
.
Simplify
3 +1
2 3  1 (2 3  1)( 3  1)
=
3 +1
( 3 +1)( 3  1)
73 3
=
3 1
73 3
=
2
15 of 38
Working:
2

3 1

3 1
= 6  2 3 3 + 1
=7 3 3
The index laws
Contents
Using and manipulating surds
Rationalizing the denominator
The index laws
Zero and negative indices
Fractional indices
Solving equations involving indices
Examination-style questions
16 of 38
Index notation
Simplify:
a to the power of 5
a × a × a × a × a = a5
a5 has been written using index notation.
The number a is
called the base.
an
The number n is called
the index, power or
exponent.
In general:
n of these
an = a × a × a × … × a
17 of 38
Index notation
Evaluate the following:
0.62 = 0.6 × 0.6 = 0.36
34 = 3 × 3 × 3 × 3 = 81
(–5)3 = –5 × –5 × –5 = –125
When we raise a
negative number to
an odd power the
27 = 2 × 2 × 2 × 2 × 2 × 2 × 2 = 128
(–1)5 = –1 × –1 × –1 × –1 × –1 = –1
(–4)4 = –4 × –4 × –4 × –4 = 256
18 of 38
When we raise a
negative number to
an even power the
The multiplication rule
When we multiply two terms with the
same base the indices are added.
For example:
a4 × a2 = (a × a × a × a) × (a × a)
=a×a×a×a×a×a
= a6 = a (4 + 2)
In general:
am × an = a(m + n)
19 of 38
The division rule
When we divide two terms with the
same base the indices are subtracted.
For example:
a5
÷
a2
a×a×a×a×a
=
a×a
= a3 = a (5 – 2)
2
4p6
÷
2p4
4×p×p×p×p×p×p
=
= 2p2 = 2p(6 – 4)
2×p×p×p×p
In general:
am ÷ an = a(m – n)
20 of 38
The power rule
When a term is raised to a power and the result
raised to another power, the powers are multiplied.
For example:
(y3)2 = y3 × y3
(pq2)4 = pq2 × pq2 × pq2 × pq2
= (y × y × y) × (y × y × y)
= p4 × q (2 + 2 + 2 + 2)
= y6 = y3×2
= p4 × q8
= p4q8 = p1×4q2×4
In general:
(am)n = amn
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Using index laws
22 of 38
Zero and negative indices
Contents
Using and manipulating surds
Rationalizing the denominator
The index laws
Zero and negative indices
Fractional indices
Solving equations involving indices
Examination-style questions
23 of 38
The zero index
Any number or term divided
by itself is equal to 1.
Look at the following division:
y4 ÷ y4 = 1
But using the rule that xm ÷ xn = x(m – n)
y4 ÷ y4 = y(4 – 4) = y0
That means that
y0 = 1
In general:
a0 = 1
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(for all a ≠ 0)
Negative indices
Look at the following division:
b2
÷
b4
=
b×b
b×b×b×b
1
1
=
= 2
b×b
b
But using the rule that am ÷ an = a(m – n)
b2 ÷ b4 = b(2 – 4) = b–2
That means that
b–2 =
1
b2
In general:
a–n =
25 of 38
1
an
Negative indices
Write the following using fraction notation:
26 of 38
1
u
This is the
reciprocal of u.
1)
u–1 =
2)
2n–4 =
2
n4
3) x2y–3 =
x2
y3
4) 5a(3 –
b)–2 =
5a
(3  b)2
Negative indices
Write the following using negative indices:
2
1) = 2a–1
a
x3
2) 4 = x3y–4
y
p2
3)
= p2(q + 2)–1
q+2
3m
4) 2
= 3m(n2 – 5)–3
3
(n  5)
27 of 38
Fractional indices
Contents
Using and manipulating surds
Rationalizing the denominator
The index laws
Zero and negative indices
Fractional indices
Solving equations involving indices
Examination-style questions
28 of 38
Fractional indices
Indices can also be fractional. For example:
1
2
What is the meaning of a ?
Using the multiplication rule:
1
2
1
2
1+1
2 2
a × a =a
= a1
=a
a= a × a
But
1
2
So
a = a
29 of 38
1
2
a is the square
root of a.
Fractional indices
Similarly:
1
3
1
3
1
3
a ×a ×a = a
1+1 +1
3 3 3
= a1
=a
a = 3 a × 3 a ×3 a
But
1
3
1
3
So
a =3a
a is the cube
root of a.
In general:
1
n
a =na
30 of 38
Fractional indices
2
3
What is the meaning of a ?
2
3
We can write a as a
2 31
.
Using the rule that (am)n = amn, we can write
2
3
1
3
a  (a )  3 a 2
2
3
We can also write a as a
1 2
3
2
3
2
.
1
3
a  (a )2  ( 3 a )2
In general:
m
n
n
a = a
31 of 38
m
or
m
n
a =
 a
n
m
Fractional indices
Evaluate the following:
1) 16
5
4
2) (0.125)
5
4
4
5
16 = ( 16)
 32
3) 36
 32
(0.125) = 8
2
3
36
 32
 32
=
1
( 36 )3
= 25
= ( 3 8)2
= 32
= 22
1
= 3
6
=4
=
32 of 38
1
216
Summary of the index laws
Here is a summary of the index laws for all rational exponents:
a m × a n = a ( m+n )
m
n
a ÷a =a
m n
(a ) = a
( m n )
mn
1
a =a
a = 1 (for a  0)
0
33 of 38
a
n
1
= n
a
1
2
a = a
1
n
n
a = a
m
n
n
a = a
m
Solving equations involving indices
Contents
Using and manipulating surds
Rationalizing the denominator
The index laws
Zero and negative indices
Fractional indices
Solving equations involving indices
Examination-style questions
34 of 38
Solving equations involving indices
We can use the index laws to solve certain types of equation
involving indices. For example:
Solve the equation 25x = 1255 – x.
25x = 1255 – x
(52)x = (53)5 – x
52x = 53(5 – x)
2x = 3(5 – x)
2x = 15 – 3x
5x = 15
x=3
35 of 38
Examination-style questions
Contents
Using and manipulating surds
Rationalizing the denominator
The index laws
Zero and negative indices
Fractional indices
Solving equations involving indices
Examination-style questions
36 of 38
Examination-style question 1
6+ 3
Show that
can be written in the form a + b 2 where
6 3
a and b are integers. Hence find the values of a and b.
Multiplying top and bottom by 6 + 3 gives



3 
6+ 3
6
6+
So
37 of 38
 = 6+2 6 3 +3
63
3
6+ 3
9 + 2 18
18 can be written as 3 2.
=
3
9+6 2
=
3
=3+2 2
a = 3 and b = 2
Examination-style question 2
a) Express 32x in the form 2ax where a is an integer to be
determined.
x
32 = 2
x2
32 = 25
So
32x = (25)x
Using the rule that (am)n = amn
32x = 25x
b) Using the answer from part a) this equation can be written as
5x
x2
2 =2
5x = x2
5x – x2 = 0
x (5 – x) = 0
x = 0 or x = 5
a)
38 of 38