maths foundation

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Starter: Test Yourself
1
What are the values of ?
Do you notice any pattern?
Can you guess what the values of are?
Can you create a rule to simplify ?
What about ?

2. Pure Maths
Topic 3
Summary Notes:
Indices

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3
OBJECTIVES
By the end of this topic you should be able to:
• Apply the laws of indices
• Use negative, zero and fractional indices
• Simplify expressions involving indices
• Solve equations involving indices
TEXT BOOK
STUDY
Textbook: Edexcel AS & A-Level Mathematics Student
Textbook - Pure Mathematics Year 1/AS
Chapter 3. Algebra: 3.2 Laws of Indices
KEY WORDS
AND PHRASES
Index, Indices, Powers
Indices

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What is an index?
4
Definition:
An index, or a power, indicates the number of times a quantity is
repeatedly multiplied by itself.
The plural form of index is indices.
Here is a list of powers of 2:
Where are these used in real-life?
Binary is used to simplify information given to a computer and this is
based on expansions of numbers into powers of 2.

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5
Laws of Indices
Part 1 Try this out on MyMaths
https://app.mymaths.co.uk/
596-lesson/indices-part-1
Part 1 #1,2,3,5,6,7
Example:
(2x5
y2
)(3y2
)3
use the power rule
Multiply the coefficients
(2 x 27)
Use the multiplication rule
y2
x y6
= 54x5
y8
= (2x5
y2
)(27y6
)

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6
Laws of Indices
Part 2 Try this out on MyMaths
https://app.mymaths.co.uk/
597-lesson/indices-part-2
Part 2 #1-7
Challenge Question:
Example:
(25 )
3
2
− (16 )
−
1
4
¿ √25
3
−
1
4
√16
25
3
2
=( √25)
3
=√253
Which is easier to evaluate?
124.5

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7
Pairing
Activity
© dfe N12 Using Indices
Match up cards from the left with cards from the right.

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The Laws of Indices
8
Rule
#
Rule What if we have more
than 2 terms?
1
2
3
4
5
6
7
8

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Additional Notes
9
Example:
Example:
Remember
Remember
“Kiss and Flip”

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10
Test Yourself
1. Evaluate
√( 8
27 )
− 2
3
2. Simplify ¿ ¿
¿
9
4
Can you fill in the
missing steps?
¿
9
16 𝑎2
𝑥2
Without using a calculator:
3. If , express in terms of .

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What happens in
Case 1 if ?
How to solve equations with indices?
11
Consider the equation
where n is a positive integer and .
Case 1: is odd
The ONLY SOLUTION to the equation is
Case 2: is even
There are TWO SOLUTIONS to the equation
NOT POSSIBLE (no real solutions)
What happens in
Case 2 if ?
Example:

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Worked Example 1:
Solving equations with an unknown index
Find the value of such that
12
Step 1: Find a common base
and rewrite all terms in this
base
Step 2: Equate indices
Step 3: Solve 𝑥=4

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13
Worked Example 2:
Solving equations with an unknown index
Following the steps on the previous slide,
1)
2)
3)

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14
Worked Examples 3
𝑎𝑚
×𝑎𝑛
=𝑎𝑚+𝑛
(𝑎𝑚
)𝑛
=𝑎𝑚 ×𝑛
Remember
What do you already know
that might be useful here?

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Simplifying Exponents with Different Bases
15
When we multiply or divide exponents with different bases there are 2 cases:
1. Simplifying Exponents with different bases and the same power.
2. Simplifying Exponents with different bases and different powers.
𝑎𝑚
×𝑏𝑚
=(𝑎𝑏)𝑚
𝑎𝑚
÷ 𝑏𝑚
=(𝑎 ÷ 𝑏)𝑚
24
× 34
=(6)4
103
2
3
=5
3
23
×52
=8x25=200
Simplify the terms separately and then
apply the required arithmetic operation.
Why it works:

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Simplifying Exponents with Different Bases
16
Evaluate:
1.
2.
Simplify:
1.
2.
What should you do first to solve this problem? Why?

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End of Topic - Indices
What questions do you have?
It is time to start practising.
Worksheet B
Seminar sheet Q3, Q5, Q7
Worksheet C Q 7, 9, 11, 13, 15
17

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Learning Check: Can you solve the following sample
exam questions?
18
1) Express the following in the form where is rational
2) Solve the following equation for
3) SHOW that the only solution to
is .

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19
OBJECTIVES
Are you able to:
• Apply the laws of indices
• Use negative, zero and fractional indices
• Simplify expressions involving indices
• Solve equations involving indices
KEY WORDS
AND
PHRASES
Index, Indices, Powers
End of Topic - Indices

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Key Formulae
20
𝑎𝑚
×𝑏𝑚
=(𝑎𝑏)𝑚
𝑎𝑚
÷ 𝑏𝑚
=(𝑎 ÷ 𝑏)𝑚