Radians• A Recap of last lesson’s work.

IndicesAims...

Revise and Use Index LawsUnderstand the effect of negative indices.

Use index laws to solve complicated indices problems

Learning Outcomes

• Name: To say what the index laws are.• Describe: How to evaluate indices.

How you use the index laws to simplify indices and expressions. How you can solve an equation with an unknown power using index laws.

• Explain: The effect of 0 or negative powers. How you can evaluate expressions that have indices in.

Independent Study

• Core 1 – Exercise 7B question 4 also 6 to 10.• Next lesson we look at fractional indices see

GCSE notes and/or mymaths lessons to get ahead.

• The Bakhshali Manuscript is an Ancient Indian (Modern day Pakistani) mathematical artefact. A leaf of birch bark found in a farmers field. Thought to date somewhere between 200-400 CE it contains the first evidence of the laws of indices.

Simplifying Indices

• You are going to be shown some problems that involve indices and how they can be simplified.

• We will see if you can spot the patterns involved.

• Are you ready? Lets go.

What Is…

a2 x a4

a6

What Is…

b5 x b6

b11

What Is…

c7 x c8

c15

What Is…

y15 x y21

y36

What Is…

z7 x z-5

z2

What Is…

n-3 x n-4

n-7

What Is…

an x am

an+m

What Is The First Index Law

• If you have two powers of the same base (number) that are multiplied this can be simplified by adding the powers.

• For indices with the same base...• an x am

= an+m

• There are more; lets try something a little different.

1 minute

What Is…

a6 ÷ a2

a4

What Is…

b12 ÷ b3

b9

What Is…

c16 ÷ c2

c14

What Is…

f3 ÷ f9

f-6

What Is…

y-4 ÷ y-9

y5

What Is…

an ÷ am

an-m

What Is The Second Index Law

• If you have two powers of the same base (number) that are divided this can be simplified by subtracting the powers.

• For indices with the same base...• an ÷ am

= an-m

• There is one more which is an extension of the first…

• Like before one more time.

1 minute

What Is… Think a5 x a5

(a5)2

a10

What Is…

(b8)3

b24

What Is…

(c2)4

c8

What Is…

(y13)13

y169

What Is…

(an)m

anm

What is the Third Index Law?

• Power of a power means the powers are multiplied.

• Using the first rule we can see this...• (an)m an x an x an ... an so that

there is m lots of an or an+n+n… which is anm

To round up…• Multiply... Add powers• Divide... Subtract powers• Power of powers... Multiply Powers

1 minute

Simplifying Indices & Algebra

• When we simplify multiplications; divisions and powers of terms that include indices we often need to apply the index laws.

• It is important that the index laws are applied correctly.

• The recall of orders of operations is also important.

Exercises• Examples• Simplify...a)3a3 x (5a4)2

b) (3a2b4)3 ÷ (9a3b) Write the following in the form 2n

9

33

2

48

Time for an activity.

What About Other Powers?

• Zero/Negative Powers...

• Continue the dividing pattern (use fractions)• a0 =

• a-n =

2-3 2-2 2-1 20 21 22 23

2 4 8

3-3 3-2 3-1 30 31 32 33

3 9 27

1 minute

What This Means

• Any powers on the bottom of a fraction can be moved into the numerator as negative powers (we usually leave the numbers as a single multiplier).

• E.g.

• It is important to use this later on in AS and A Level!

323

2

2

1

12

6 bab

a

1 minute

Examples using negative indices

• Evaluate...• (1/3)-3

• x-3 if x = 2

?4 if 2

5 isWhat

2 x

x

Time for part 2.

Still One Topic To Do

• Lest you think these really easy here are some past paper questions…

NOT DONE ANY LIKE THIS YET

How?

Time to put it all together.

1 minute

Learning Outcomes

• Name: To say what the index laws are.• Describe: How to evaluate indices.

How you use the index laws to simplify indices and expressions. How you can solve an equation with an unknown power using index laws.

• Explain: The effect of 0 or negative powers. How you can evaluate expressions that have indices in.

Learning Outcomes• Name: There are three index Laws anxam=an+m;

an÷am=an+m; (an)m=anm

• Describe: Indices represent repeated multiplication. By applying the index laws to a problem with powers of the same unknowns you can simplify the expression. To solve equations with unknown powers you can write both sides as powers of one number then make the powers equal to each other and solve.

• Explain: Any value to the power of 0 is 1. Any negative power a-n is 1 over the number to the positive power 1/a

n. As with all algebra what works with numbers will work with expressions and indices are no different; you can substitute in powers or bases. E.g. xy if x=2 and y = 5 means xy=25=32