c1: chapter 1 – algebraic manipulation dr j frost ([email protected]) last modified: 2...
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C1: Chapter 1 – Algebraic Manipulation
Dr J Frost ([email protected])
Last modified: 2nd September 2013
Starter
Expand the following.
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Recap: Basic Laws of Indices
𝑎3×𝑎8=𝑎11 (𝑎3 )8=𝑎24
71=7 𝑎0=1
2− 3=1
23=18 64
13= 3√64=4
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√𝑎=𝑎12 𝑏4
𝑏−1=𝑏5? ?
‘Flip Root Power’ method
32− 35 �⃗�𝑙𝑖𝑝 1
3235
�⃗�𝑜𝑜𝑡 123�⃗�𝑜𝑤𝑒𝑟 1
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More examples
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Textbook Fail
Example 7d on page 9 is wrong:
25− 32= 1125
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Whenever you have fractional powers where the denominator is even, by DEFINITION, you only consider the positive solution.
Exercises
3𝑎6×𝑎7=3𝑎13
2𝑏7×3𝑏−2=6𝑏51
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3
𝝅
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2 (𝑐3 )2=2𝑐6
𝑑7
𝑑3=𝑑4
𝑓 4
𝑓 −2= 𝑓 6
5 ( (𝑚2 )3 )4=𝑚24
6 𝑎√𝑎=𝑎32
7𝑎2
√𝑎=𝑎
32
8(𝑏3 )4
(√𝑏)20=𝑏2
9
Simplify:
Evaluate:
90.5=3810.25=33−3=
127
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11
12 2− 0.5=
1
√213 8
23=4
14 432=8
15 9− 32= 127
16 8− 13=12
17 64− 23= 116
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Skill 2: Power of a product
(2𝑎𝑏 )5=32𝑎5𝑏5
(𝑎3𝑏2𝑐 )2=𝑎6𝑏4𝑐2
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Skill 2b: Power of a fraction
( 38 )2
= 964
( 75 )−1
=57
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( 23 )−2
= 94
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‘Flip Root Power’ method
( 278 )−23 �⃗�𝑙𝑖𝑝( 827 )
23 �⃗�𝑜𝑜𝑡 ( 23 )
2
�⃗�𝑜𝑤𝑒𝑟 49
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Exercises
(2𝑎𝑏2 )3=8 𝑎3𝑏6
(9𝑎2 )12=3𝑎
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(16 𝑎4𝑏3 )12=4 𝑎2𝑏
32
(27 𝑎9𝑏4 )13=3 𝑎3𝑏
43
Simplify:
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(8𝑎6𝑏12)23=4𝑎4𝑏8
6 (16 𝑎6𝑏12)32=64 𝑎9𝑏18
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7
( 56 )−1
=65
9 ( 23 )−3
=278
10
( 94 )12=32
11 ( 6427 )−13=34
Evaluate:
( 916 )−32=6427
12
8 ( 23 )3
= 827??
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Skill 3: Changing Base
Express in the form , stating in terms of .
Solve
Express in the form , stating in terms of .
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FactorisingSKILL 1
Taking out a single factorSKILL 2
SKILL 3Difference of two squares
SKILL 4
(Use ‘commando method’ or ‘splitting the middle term’ method)
𝑥2−7 𝑥−30=(𝑥+3 ) (𝑥−10 )??
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Exercises
Page 7 – Exercise 1EEvens
Page 9 – Exercise 1F1g,h,i , 2
Recap
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And that’s it!
Laws of Surds
Using these laws, simplify the following:
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Laws of Surds
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Work these out with neighbour. Simplify as much as possible.We’ll feed back in a few minutes.
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Expansion involving Surds
(2+√8 ) (7−√2 )=10+12√2?
It’s convention that the number inside the surd is as small as possible, or the expression as simple as possible. This sometimes helps us to further manipulate larger expressions.
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Simplifying Surds
This sometimes helps us to further manipulate larger expressions.
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Simplifying Surds
Here’s a surd. What could we multiply it by such that it’s no longer an irrational number?
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Rationalising Denominators
In this fraction, the denominator is irrational. ‘Rationalising the denominator’ means making the denominator a rational number.
What could we multiply this fraction by to both rationalise the denominator, but leave the value of the fraction unchanged?
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There’s two reasons why we might want to do this:1. For aesthetic reasons, it makes more sense to say “half of root 2” rather
than “one root two-th of 1”. It’s nice to divide by something whole!2. It makes it easier for us to add expressions involving surds.
Rationalising Denominators
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Rationalising Denominators
What is the value of the following. What is significant about the result?
This would suggest we can use the difference of two squares to rationalise certain expressions.
What would we multiply the following by to make it rational?
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Rationalising Denominators
Quickfire DO2S!
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5√6−2
=5√6−52
2√7+√3
=2√5−2√34
Rationalise the denominator. Think what we need to multiply the fraction by, without changing the value of the fraction.
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Rationalising Denominators
√2√5−√3
=√10+√62
√5−3√2+3
=− √10−3√5−3√2+97
Rationalise the denominator. Think what we need to multiply the fraction by, without changing the value of the fraction.
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Rationalising Denominators
Exercises
Page 12 – Exercise 1HOdds