Year 8 Laws of Indices Dr J Frost Last modified: 29 th August 2015

“Power” or “index” (plural: “indices”) or “exponent” 3434 ? “Base” ? The whole expression is sometimes (confusingly) referred to as a ‘power’ or ‘power expression’. Terminology Recap   We say this as “3 to the power of 4” or “3 raised to the power of 4” or “3 to the 4”.

3 5 × 3 4 How would I write this multiplication out in full? Therefore, how could I write the result of this multiplication in the form 3 k ? 3939 ? Multiplying Power Expressions with the Same Base

a b × a c = a b+c ? 1 st Law of Indices 

How would I write this multiplication out in full? Therefore, how could I write the result of this multiplication in the form 4 k ? 4545 ? Dividing power expressions with the same base

abacabac = a b-c ? 2 nd Law of Indices 

(4 2 ) 3 How would I write this multiplication out in full? Therefore, how could I write the result of this multiplication in the form 4 k ? 4646 ? Raising a power to a power 

(a b ) c = a bc ? 3 rd Law of Indices 

3 -1 At this point, it doesn’t make sense to say “Multiply 3 by itself negative 1 times”. We’ll have to use a different approach! 3 3 = = = = = 3 -2 = ? ? Is there a pattern we can see that will help us out? Zero and negative indices ?

a 0 = 1 a -b = a 1 = a 1ab1ab Final Laws of Indices 

Mastermind Occupation: Student Favourite Teacher: Dr Frost Specialist Subject: Laws of Indices

Instructions: Everyone starts by standing up. You’ll get a question with a time limit to answer. If you run out of time or get the question wrong, you sit down. The winner is the last man standing. Warmup: 2 3 × 2 4 = 2 7 ? Start Question > (2 3 ) 4 = 2 12 ? Start Question > = 2 3 ? Start Question >

_1_ × 4 3 = 4 10 ? Start Question > (3 5 ) 2 = 3 10 ? Start Question > = 9 9 ? Start Question > 7 4 × 7 6 = 7 10 ? Start Question > = 5 4 ? Start Question > (4 6 ) 3 = 4 18 ? Start Question > (2 2 ) 2 = 2 4 ? Start Question > 2 -1 = ? Start Question > abc d e f g h

_1_ 8 _1_ × 7 -2 = 7 5 ? Start Question > (5 3 ) -2 = 5 -6 ? Start Question > = 10 3 ? Start Question > 8 -2 × 8 4 = 8 2 ? Start Question > = 8 9 ? Start Question > 2 -3 = ? Start Question > 5 0 = 1 ? Start Question > 4 -2 = ? Start Question > abc d e f g h

_1_ 5 6 _1_ × 4 -2 = 4 -4 ? Start Question > (3 -2 ) -2 = 3 4 ? Start Question > 9 -2 = 9 0 = 1 ? Start Question > 1 4 × 1 6 = 1 10 = 1 ? Start Question > = 10 4 ? Start Question > (5 -3 ) 2 =5 -6 = ? Start Question > 3 -3 = ? Start Question > abc d e g h INSTANT DEATH Start Question > f

_1_ = 1 ? Start Question > (3 0 ) 2 = 1 ? Start Question > = 5 2 ? Start Question > (2 4 × 2 6 ) 2 = 2 20 ? Start Question > 5 1 x 5 2 x 5 3 = 5 6 ? Start Question > ((4 1 ) 2 ) 3 = 4 6 ? Start Question > (2 3 × 2 3 ) 3 = 2 18 ? Start Question > 3 -4 = ? Start Question > abc d e f g h

4 7 × (3 5 ) 43 a = 4 8 ? Start Question > = 3 17 ? Start Question > b (7 3 ) 3 (7 2 ) 3 = 7 3 ? Start Question > c 5 8 × × 5 -1 ((3 2 ) 2 ) = 5 16 ? Start Question > = 3 6 ? Start Question > e (7 1 ) 3 (7 2 ) 1 ×7 4 = 7 5 ? Start Question > fd

? ? ? ? Challenges ?

Exercise 1 Simplify the following. 1 Please ensure you write out the question. 2 Simplify the following. 3 Evaluate the following (i.e. give as a fraction or integer with no power) 4 Simplify the following. 11 22 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?

(Hint: can we express 9 as a power of 3 perhaps?) ? Changes of Base Solve the following equation. Express as a single power. ? The strategy therefore is to find what both bases are a power of (e.g. 4 and 8 are both powers of 2), and replace them as such.

A few more examples… ?? Express as a single power of 3: ? Solve

2 8 = 4 x x = = 9 x x = 6 ? x = 4 ? 8 x = 4 12 ? x = x = ? Test Your Understanding Express as a single power: ? ?

Exercise 2 Please ensure you write out the question. Solve the following. Express as a single power  ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?