GCSE Laws of Indices Dr J Frost jfrosttiffin

GCSE : : Laws of Indices Dr J Frost (jfrost@tiffin. kingston. sch. uk) @Dr. Frost. Maths Objectives: (a) Understand laws for multiplying power expressions, raising a power to a power and dealing with 0 or negative exponents. GCSE: (b) Deal with fractional exponents. (c) Deal with problems involving a mixture of bases. Last modified: 9 th July 2021

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Terminology ! ? “power” “exponent” or “index”? (plural “indices”) “Base” ? The exponent tells us how many times the base appears in a product. ! We say this as “ 3 to the power of 4” or “ 3 raised to the power of 4” or “ 3 to the 4”.

More on notation Phrase “Powers of 2. ” Precise meaning ? “ 3 raised to the power of 2. ” ?

Understanding powers When the exponent is a positive integer (whole number), it indicates how many times the base is repeated in the multiplication. 5 appears 2 times. ? ? ? It’s possible for the exponent to be fractional, 0 or negative. We’ll deal with these later!

Multiplying powers How would we simplify this?

Quickfire Questions Your teacher will target various people. Do in your head! ? ? ? ? ?

Dividing Powers How would we simplify this? Remember that we can simplify fractions by dividing the numerator and denominator by the same number (or term).

Quickfire Questions Your teacher will target various people. Do in your head! ? ? ?

Raising a Power to a Power How would we simplify this?

Quickfire Questions Your teacher will target various people. Do in your head! ? ? ? ?

Mastermind Occupation: Student Insert your ame r’s n e h c a Favourite Teacher: Drtheer. Frost e 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. Start Warm 3 4 7 3 4 = ? 212 2 Question × 2 = ? 2 (2 ) Question > > up: 26 Start ? 3 = 2 3 > 2 Question

a b 47 ×Start 43 = ? 410 Start 5 (3 )2 = ? 310 Question > d 57 Start 4 = 5 ? 3 5 Question > c Question > e Start 4 7 × 76 =? 710 Question > g Start 2)2 = ? 24 (2 Question > f 91 Start? 9 1 = 9 Question > 92 Start 6 )3 = ? (4 Question > 418

a b 77 ×Start 7 -2 =? 75 Start 3 -2 = ? -6 (5 ) 5 Question > d 87 Start? 9 = 8 -2 8 Question > c e Start -2 8 × 84 =? 82 f 2 Start _1_ -3 2 =? Question > g Start 2 x p=? p Question > p 3 10 Start? 3 5 = 10 Question > 10 Question 8 > h x 9 Start 1 ? x = Question x-3 2 >

a Start -2 × 4 -2 = ? > 4 4 Question b -4 d 101 Start 10 ? 4 = Question > 10 3 Start -2)-2 = ? 3 Question >4 (3 e Start 4 1 × 16 =? 110 Question > =1 g Start -3)2 =5 -6 ? > (5 Question c f 9 -2 Start? 0 =9 = Question > -2 9 1 Start Question > INSTANT DEATH

a 50 Start? 2 =5 > -2 Question 5 b 51 x. Start 52 x 53 Question =? 56 > e Start 1 ((4 )2)3 =? Question > 46 c Start Question 4 (2 × 2>6)2 =? 220 f Start 3 × 2 3 )3 = (2 Question >? 218

a 47 ×Start 43 ? = Question > 2 4 48 d 58 ×Start 58 ? = Question > 51 × 5 - 16 5 1 b c (35 Start ) = 3? 17 (73 Start ) = 7? 3 4 Question > e 33 ((32 Start )2)2= 3? 6 Question > 2 3 3 Question > f (72) 3 (71 Start ) × 74 =? 3 Question > 2 75 (7 ) 1

Group Challenges 1 2 ? 3 ? 4 ? ? N ?

Exercise 1 a 1 Simplify the following. a ? b ? c ? d ? e ? f ? g ? h i j k l m n o ? ? ? ? Questions on provided worksheet. 2 Simplify the following. a b ? c d ? g 4 ? ? a b c ? e f ? ? 3 d ? e ? ? ? Exercise continues on next slide… ? ?

Exercise 1 a Questions on provided worksheet. 5 N 1 ? ? ? 6 ? N 2 Simplify the following. ? ?

Zero and negative indices Is there a pattern we can see that will help us out? At this point, it doesn’t make sense to say “The product of -1 threes”. We’ll have to use a different approach! 33 = 27 2 3 =9 1 3 =3 30 = 1? 3 -1 = 1 ? 3 3 -2 = 19?

Quickfire Questions Your teacher will target various people. Do in your head! ? ? ? ? ? ? A power of -1 therefore ‘flips’ (reciprocates) the fraction. ?

Mini Exercise (Exercise 1 b) 1 Questions on provided worksheet. N ? a ? ? b c ? d 2 ? ?

A reminder of the Laws of Indices ? ? ? ?

Fractional Indices And how could we prove this? ?

Fractional Indices ? ?

Examples ? ? ? ? ? ? ?

Test Your Understanding So Far… ? ? ?

What if the numerator is not 1? ‘Workings’-wise I usually skip his step. ? …then just deal with what’s left. ? Using denominator, do 5 th power of 32 first to get 2 (but still have numerator left in the power to deal with) Best to deal with negative in power first. Recall this does “ 1 over” the expression without the minus. ?

A few more examples ? Recall that “reciprocating” a fraction will cause it to flip. ? ?

Test Your Understanding ? ? ? ?

Exercise 2 Questions on provided worksheet. 1 ? 2 ? 8 5 ? 10 ? ? ? 14 ? ? 15 6 13 ? 9 4 12 ? ? 3 11 ? 7 ? Write the following expression without using indices: ? ?

Applying indices to products ? ?

Examples ? ?

Test Your Understanding ? ?

Exercise 3 1 2 3 4 5 6 7 8 9 10 11 Questions on provided worksheet. ? ? ?

Law of Indices Backwards This part of the topic is a bit more Further Mathsey… The ‘thinking backwards’ method ? The ‘cancelling the power’ method. ?

Further Examples ? ?

Test Your Understanding ? ?

Exercise 4 Questions on provided worksheet. ? ? 1 2 6 ? 3 7 ? ? 4 ? ? ? 5 ? ?

Changing bases ?

Changing bases 1 2 First convert everything to powers of 2. ? First convert everything to powers of 2… ? ? ? 3 ?

Test Your Understanding 1 ? 2 ?

Exercise 5 1 ? a b ? c d e f g ? 4 ? ? ? a b c d e 5 ? ? 2 3 ? N ? ? ? ?