GCSE :: Laws of Indices Dr J Frost (jfrost@tiffin.kingston.sch.uk) @DrFrostMaths 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: 9th July 2021 www.drfrostmaths.com Everything is completely free. Why not register? Register now to interactively practise questions on this topic, including past paper questions and extension questions (including UKMT). Teachers: you can create student accounts (or students can register themselves), to set work, monitor progress and even create worksheets. With questions by: Dashboard with points, trophies, notifications and student progress. Teaching videos with topic tests to check understanding. Questions organised by topic, difficulty and past paper. Terminology ! ? “power” 4 “exponent” or “index”? (plural “indices”) 3 =3×3×3×3 “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 3 4 You may have heard the 4 referred to as the power. But ‘power’ refers to the whole expression 34 ; the 4 is the exponent! Phrase “Powers of 2.” “3 raised to the power of 2.” Precise meaning Powers where the base is 2: 3 , 24 21 , 22 , 2? 32 . ‘Raised’ here means ‘turned into a power’. While “power of 2” might suggest ? the 2 is the ‘power’, it’s really short for “power with an exponent 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. 52 = 𝟓 × 𝟓 = 𝟐𝟓 ? 23 = 𝟐 × 𝟐 × 𝟐 = ?𝟖 34 = 𝟑 × 𝟑 × 𝟑 × ?𝟑 = 𝟖𝟏 25 = 𝟐 × 𝟐 × 𝟐 × ?𝟐 × 𝟐 = 𝟑𝟐 131 = 𝟏𝟑 ? It’s possible for the exponent to be fractional, 0 or negative. We’ll deal with these later! Warning: Sometimes people incorrectly describe “43 ” as “4 multiplied by itself 3 times”. This would suggest there are 3 multiplications, but 4 × 4 × 4 actually only has 2 multiplications! Multiplying powers How would we simplify this? 3 𝑥 × 2 𝑥 𝑥 3 means 3 𝑥’s multiplied together. =𝑥×𝑥×𝑥 × 𝑥×𝑥 5 =𝑥 In total we had 5 𝑥’s multiplied together. ! 1st law of indices: 𝒙𝒂 × 𝒙𝒃 = 𝒙𝒂+𝒃 i.e. when we multiply two powers, we add the exponents. Quickfire Questions Your teacher will target various people. Do in your head! 𝑥 5 × 𝑥 4 = 𝒙?𝟗 ? 𝑦10 × 𝑦10 = 𝒚𝟐𝟎 𝑥 2 × 𝑥 3 × 𝑥 4 = 𝒙𝟗? 𝑝 × 𝑝4 = 𝒑𝟓? ? 𝑥 × 𝑥 2 × 𝑥 9 = 𝒙𝟏𝟐 ? 𝑦 𝑘 × 𝑦 2 = 𝒚𝒌+𝟐 ? 𝑝𝑎 × 𝑝𝑛 = 𝒑𝒂+𝒏 𝑞 × 𝑞𝑎 = 𝒒𝟏+𝒂 ? Fro Tip: When there is no exponent, you can raise the term to the power of 1: 𝑥 → 𝑥1 Dividing Powers How would we simplify this? 5 𝑥 𝑥3 𝑥×𝑥×𝑥×𝑥×𝑥 = 𝑥×𝑥×𝑥 2 =𝑥 Remember that we can simplify fractions by dividing the numerator and denominator by the same number (or term). ! 2nd law of indices: 𝒙𝒂 ÷ 𝒙𝒃 = 𝒙𝒂−𝒃 i.e. when we divide two powers, we subtract the exponents. Quickfire Questions Your teacher will target various people. Do in your head! ? 2100 ÷ 22 = 𝟐𝟗𝟖 𝑥 10 𝟕? = 𝒙 𝑥3 𝑦 20 𝟐𝟏 ? = 𝒚 −1 𝑦 𝑥 15 ÷ 𝑥 = 𝒙𝟏𝟒? 𝑥 3 × 𝑥 3 𝒙𝟔 𝟒 = = 𝒙 ? 𝑥2 𝒙𝟐 Raising a Power to a Power How would we simplify this? 3 4 𝑥 3 𝑥 = × 12 =𝑥 3 𝑥 × 3 𝑥 ! 3rd law of indices: 𝒙𝒂 𝒃 = 𝒙𝒂𝒃 i.e. when we raise a power to a power, we multiply the exponents. × 3 𝑥 Quickfire Questions Your teacher will target various people. Do in your head! 𝑦 3 5 = 𝒚𝟏𝟓? 𝑥 3 × 𝑥 5 = 𝒙𝟖 ? 𝑝7 8 = 𝒑𝟓𝟔? 𝑞 6 × 𝑞 9 = 𝒒𝟏𝟓? 𝑚10 4 = 𝒎𝟒𝟎? 𝑢8 × 𝑢10 = 𝒖𝟏𝟖? −𝑏 −𝑐 𝑎 = 𝒂𝒃𝒄? 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: 3 × 2Question 4 = 27? > 2Start 26 3? Start Question > = 2 3 2 4 = 2?12 > Start (23)Question a b 7 × 43 = 410 4Start Question? > d 57 c 5)2Question Start (3 = 3?10 > e f 4 × 76 = 710 7Start Question? > 4? = 5 Start Question > 3 5 911 9? = 9 Start Question > 2 9 g Start 2 = 2?4 > (22)Question 3 = 418 Start (46)Question ?> a b 7 × 7-2 = 75? 7Start Question > d 87 c 3)-2Question Start (5 = 5-6? > e f -2 × 84 = 8?2 8Start Question > ?9 > = 8 Start Question -2 8 Start p2 xQuestion p= p3? > _1_ -3 Start ? > 2 =Question 8 h g 105 ?3 > = 10 Start Question 102 9 x 12 Start Question ? > x = -3 x a b -2 ×Question 4Start 4-2 = 4? >-4 d 101 c Start -2 Question -2 4? > (3 ) = 3 e f 4 × 1Question 6 = 110?=> 1 1Start Start=Question 10? 4 > 10-3 -2 9 Start Question 0?= 1> = 9 9-2 Start Question > INSTANT DEATH g -3)2 Question (5Start =5-6 ? > a 0 5 2? Start = Question > 5 5-2 b 1 x 5 2 x 53 5 Start Question > = 5?6 e c 4×2 6)2 = 2>?20 (2Start Question f 1)2Question Start ((4 )3 = 4?6 > 3 × Question (2Start 23)3 = 2>?18 a d 7 × 43 4Start Question ?8> = 4 42 8 × 58 5 Start Question ?>16 = 5 51 × 5-1 b e 5)4 (3 Start Question ?17 > = 3 33 2)2)2 ((3 Start Question ?6> = 3 32 c f 3)3 (7 Start Question ?3 > = 7 (72)3 1)3 (7 4=7 Start Question ?>5 ×7 (72)1 Group Challenges 1 What is half of 27 ? 𝟕 𝟕 2 𝟐 𝟐 = ?𝟏 = 𝟐𝟔 𝟐 𝟐 3 What is a quarter of 4𝑥 ? What is a ninth of 399 ? 𝟗𝟗 𝟗𝟗 𝟑 𝟑 = ?𝟐 = 𝟑𝟗𝟕 𝟗 𝟑 4 What is the square 8 root of 3 ? 𝟒𝒙 𝟒𝒙 = 𝟏? = 𝟒𝒙−𝟏 𝟒 𝟒 N 4𝑥 + 4𝑥 + 4𝑥 + 4𝑥 = 416 𝟑𝟒 𝒃𝒆𝒄𝒂𝒖𝒔𝒆? 𝟑𝟒 𝟐 What is 𝑥? The LHS is 4 ∙ 4𝑥 = 41 4𝑥 = 4𝑥+1 . So 𝑥 + 1 = 16? → 𝑥 = 15 = 𝟑𝟖 Exercise 1a 1 Simplify the following. a 𝑥 2 × 𝑥 3 = 𝒙𝟓? b 𝑥 2 3 = 𝒙𝟔? 20 𝟏𝟔 c 𝑚 4 = 𝒎? 𝑚 d 𝑦 8 × 𝑦 −2 = 𝒚𝟔? e 𝑞 × 𝑞3 = 𝒒𝟒? f 𝑝5 2 = 𝒑𝟏𝟎 ? 𝟗 3 5 g 𝑥×𝑥 ×𝑥 =𝒙? 𝑦6 h 𝑦2 = 𝒚𝟒? i 𝑥 2 × 𝑥 𝑎 = 𝒙𝟐+𝒂 ? 2 𝑦 𝟐𝒚 =𝒙? j 𝑥 12 k 𝑥 3 = 𝒙𝟗? 𝑥 𝑥 12 l = 𝒙𝟏𝟏 𝑥 𝑝 m −𝑞 −𝑟 = 𝒑𝒒𝒓 ? n o 𝑤4 𝑤 −4 𝑎𝑏 𝑎−𝑏 ? = 𝒂𝟐𝒃 ? = 𝒘𝟖 ? Questions on provided worksheet. 2 Simplify the following. 𝑦 10 a 𝑦 × 𝑦 5 = 𝒚𝟔? 𝟏𝟎 b 𝑥 2 × 𝑥 3 2 = 𝒙? c d 4 𝑝12 𝑝3 𝑥 10 ×𝑥 9 𝑥5 3 e 𝑞 f 2 3 𝑥 2𝑦 𝑥𝑦 4 𝟑𝟕 × 𝑝 = 𝒑? = 𝒙𝟒? × 3 𝑞5 𝑞 2 =𝒒? 𝟏𝟒 = 𝒙𝟑𝒚 ? g 𝑥 × 𝑥 × 𝑥3 2𝑥 ×2𝑦 3 If 3 2 of 𝑦? 𝟏𝟓 = 𝒙? ? Simplify the following: 𝒚𝟓 a 2𝑥 2 𝑦 × 3𝑥𝑦 4 = 𝟔𝒙𝟑? b 5𝑝3 𝑞 4 × 5𝑝𝑞 = 𝟐𝟓𝒑?𝟒 𝒒𝟓 10 c 10𝑥 𝑦 = 𝟓𝒙𝟗? d e 2 2 = 27 , what is 𝑥 in terms 𝒙 = 𝟏𝟎 − 𝒚 2𝑥𝑦 36𝑘 3 𝑚4 𝟔 −𝟐 𝟑 = 𝒌 𝒎 30𝑘 5 𝑚 𝟓 2𝑥𝑦×2𝑥 10 𝟏 −𝟗 = 𝒙 𝒚 8𝑥 20 𝟐 ? ? Exercise continues on next slide… Exercise 1a 5 6 Questions on provided worksheet. [Edexcel GCSE June2003-6H Q17a] If 𝑥 = 2𝑝 , 𝑦 = 2𝑞 , express the following in terms of 𝑥 and/or 𝑦: (i) 2𝑝+𝑞 = 𝟐𝐩 × 𝟐𝒒 = 𝒙𝒚 (ii) 22𝑞 = 𝟐𝒒 𝟐 = 𝒙𝟐 N1 Solve (iii) 2𝑝−1 = N2 Given that 𝑥 12 = 𝑥, 𝟐𝒑 𝟐𝟏 ? ?𝒙 =? 𝟐 Simplify the following. a) 3𝑦 + 3𝑦 + 3𝑦 = 3𝑦+1 ? 2𝑦 2𝑦 2𝑦+1 b) 2 + 2 = 2 ? c) 2𝑥 + 2𝑥 2 = 22𝑥+2 ? 2𝑥 5 23 = ? 𝒙= 2 24 𝑥 𝟒 𝟗 express 4𝑥 + 4𝑥 as a single power of 4. 𝟒𝒙 + 𝟒𝒙 = 𝟐 × 𝟒𝒙 = 𝟒 × 𝟒𝒙 = 𝟏 𝟒𝟐 =𝟒 ? × 𝟒𝒙 𝒙+ 𝟏 𝟐 Zero and negative indices Is there a pattern we can see that will help us out? 0 3 −1 3 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? ÷3 ÷3 ÷3 Quickfire Questions Your teacher will target various people. Do in your head! 𝟏 −2 𝟏 6 = ? 𝟑𝟔 ? 4−1 = −1 𝟒 2 𝟐 𝟑 ? = 𝟏 ÷ = 𝟏 3 𝟑 𝟐 ? 5−1 = 4 −1 𝟓 𝟓 = ? 60 = 𝟏 ? 5 𝟒 −1 3 𝟖 𝟏 𝟏 −2 = ? 7 = 𝟐 =? 8 𝟑 𝟕 𝟒𝟗 𝟏 𝟏 −2 ? 8 = 𝟐= 𝟖 𝟔𝟒 𝟏 𝟏 −3 ? 2 = 𝟑= 𝟐 𝟖 0 3 =𝟏? 𝟏 −1 4 = ? 𝟒 𝟏 −3 5 = 𝟑 =?𝟏𝟐𝟓 𝟓 1 2 −1 3 5 −2 𝟑 =𝟏÷ 𝟓 2 3 −2 𝟑 = 𝟐 5 7 −2 3 4 −3 =𝟐 A power of -1 therefore ‘flips’ (reciprocates) the fraction. ? 𝟐 𝟐 𝟗 =? 𝟒 = 𝟒𝟗 𝟐𝟓 ? = 𝟔𝟒 𝟐𝟕 ? ?÷ =𝟏 𝟗 𝟐𝟓 = 𝟐𝟓 𝟗 Mini Exercise (Exercise 1b) 1 Determine the value of: 𝟏 a 6−1 = ? 𝟔 b 90 = 𝟏 ? 𝟏 c 8−2 = 𝟔𝟒 ? −2 8 𝟖𝟏 d = ? 9 𝟔𝟒 Questions on provided worksheet. N [Edexcel GCSE June2003-6H Q17b] Let 𝑥 = 2𝑝 , 𝑦 = 2𝑞 , If 𝑥𝑦 = 32 and 2𝑥𝑦 2 = 32, find the value of 𝑝 and the value of 𝑞 . Dividing the second equation by first: 𝟏 𝟐𝒚 = 𝟏 → 𝒚 = 𝟐 𝟑𝟐 𝟏 𝒙= = 𝟑𝟐 ? ÷ 𝟐 = 𝟔𝟒 𝒚 Therefore 𝟐𝒑 = 𝟔𝟒 → 𝟔 𝟏 𝟐𝒒 = → 𝒒 = −𝟏 𝟐 2 𝒚 = −𝟐 (as 𝟐−𝟐 = ? 𝟏 𝟐𝟐 𝟏 𝟒 = ) A reminder of the Laws of Indices ? 𝑎𝑏 × 𝑎𝑐 = 𝑎𝑏+𝑐 𝑎0 = 1 ? 𝑎𝑏 𝑏−𝑐 ? = 𝑎 𝑐 𝑎 𝑎1 = 𝑎 ? 𝑎 𝑏 𝑐 = 𝑎𝑏𝑐? −𝑏 1 = 𝑏? 𝑎 𝑎 Fractional Indices 1 𝑥2 = 𝑥 And how could we prove this? 𝒙× 𝒙=𝒙 But it’s also the case that: 𝟏 𝒙𝟐 𝟏 𝒙 ?𝟐 × by laws of indices. 𝟏 𝟐 So 𝒙 = 𝒙 = 𝒙𝟏 Fractional Indices 1 𝑥3 1 𝑥𝑛 = 3 = 𝑛 ? 𝑥 𝑥 ? Examples 1 642 1 643 1 812 1 814 ? = 𝟔𝟒 = 𝟖 = 𝟑 𝟔𝟒? = 𝟒 = 𝟖𝟏?= 𝟗 = 𝟒 𝟖𝟏? = 𝟑 0.25 16 3 𝑥 2 = 2? 1 2 𝑥 3 ?= = −1000 1 3 ? = −10 2 𝑥3 Test Your Understanding So Far… 1 ? 2 36 = 𝟑𝟔 = 𝟔 1 𝟑 ? 83 = 𝟖 = 𝟐 1 𝟓 −32 5 = −𝟑𝟐? = −𝟐 What if the numerator is not 1? 3 92 = 2 325 1 3 92 ‘Workings’-wise I usually skip his step. ? 3 = 3 = 27 …then just deal with what’s left. 2 =2 =4 3 − 16 4 ? Using denominator, do 5th power of 32 first to get 2 (but still have numerator left in the power to deal with) = 1 Best to deal with negative in power first. Recall this does “1 over” the expression without the minus. 3 4 1 1 = ?3 = 2 8 A few more examples 3 − 4 2 27 8 1 9 = 𝟏 𝟑 𝟒𝟐 2 − 3 𝟏 𝟏 = ?𝟑 = 𝟐 𝟖 𝟖 = 𝟐𝟕 1 − 2 = 𝟏 𝟗𝟐 𝟐 𝟑 =?𝟑 Recall that “reciprocating” a fraction will cause it to flip. 𝟐 ? = 𝟑 𝟐 𝟒 = 𝟗 Test Your Understanding 2 273 3 − 9 2 𝟐 ? =𝟑 =𝟗 𝟏 𝟏 𝟏 = 𝟑 = ?𝟑 = 𝟑 𝟐𝟕 𝟗𝟐 125 64 4 9 3 − 2 1 − 3 𝟔𝟒 = 𝟏𝟐𝟓 𝟗 = 𝟒 𝟑 𝟐 𝟏 𝟑 ? 𝟑 = ? 𝟐 𝟒 = 𝟓 𝟑 𝟐𝟕 = 𝟖 Exercise 2 1 1 1002 2 1 1253 3 16−0.5 4 5 6 − 27 4 83 − 8 2 3 ? = 10 = 5? 1 = ? 4 1 = ? 9 Questions on provided worksheet. 7 8 1 −3 64 2 −3 64 1 = ? 4 11 1 = ? 16 9 2 325 10 3 −5 32 = 4? 1 = ? 8 12 13 14 64 27 9 16 16 81 8 27 1 −3 3 −2 3 −4 5 −3 3 =? 4 64 = ? 27 𝟐𝟕 = ? 𝟖 𝟐𝟒𝟑 = ? 𝟑𝟐 = 16? 1 3 1 = ? 2 15 Write the following expression without using indices: 𝑥 −0.5 1 = ? 𝑥 Applying indices to products 𝑎𝑏 𝑎+𝑏 2 2 2? 2 =𝑎 𝑏 = 𝑎 + 𝑏 ?𝑎 + 𝑏 2 2 = 𝑎 + 2𝑎𝑏 + 𝑏 The moral of the story: 1. Applying a power to a product applies the power to each term. 2. Applying a power to a sum does NOT apply power to each term. i.e. 𝑎 + 𝑏 𝑛 ≠ 𝑎𝑛 + 𝑏𝑛 in general. Examples 2𝑥 2 3𝑥 2 𝑦 ?𝟐 = 𝟒𝒙 3 1 6 9𝑥 2 6 8𝑥 𝑦 = 𝟐𝟕𝒙?𝟔 𝒚𝟑 = 1 3 𝟑 ? 𝟑𝒙 = 𝟏 ?𝟐 𝟑 𝟐𝒙 𝒚 Test Your Understanding Simplify 3𝑥 2 𝑦 3 2 9𝑥 4 𝑦? 6 Simplify 9𝑥 4 𝑦 1 2 1 2 ?2 3𝑥 𝑦 Exercise 3 Questions on provided worksheet. Simplify the following: 𝟐 1 𝑥𝑦 2 = 𝒙𝟐 𝒚 ? 2 3𝑥 2 = 𝟗𝒙?𝟐 𝒚𝟒 3 𝑥𝑦 2 2 = 𝒙𝟐? 4 2𝑐𝑑 4 5 𝑎𝑏2 6 1 2 9𝑎 2 7 8 9 10 11 3 3 = 𝟖𝒄𝟑? 𝒅𝟏𝟐 = 𝒂𝟑 𝒃?𝟔 = 𝟑𝒂 ? 1 𝟑 4 3 𝟐 16𝑎 𝑏 2 = 𝟒𝒂 𝒃𝟐 1 𝟒 9 4 𝟑 27𝑎 𝑏 3 = 𝟑𝒂 𝒃𝟑 2 6 12 3 8𝑎 𝑏 = 𝟒𝒂𝟒 𝒃𝟖 3 6 12 16𝑎 𝑏 2 = 𝟔𝟒𝒂𝟗 𝒃𝟏𝟖 2 𝟏𝟎 6 5 𝟒 3 27𝑥 𝑦 = 𝟗𝒙 𝒚 𝟑 ? ? ? ? ? Law of Indices Backwards This part of the topic is a bit more Further Mathsey… 3 4 Solve 𝑥 = 27 The ‘thinking backwards’ method The ‘cancelling the power’ method. 3 If I had some number to the power of 4, what would I do to it? Find the 4th root then cube it. So going backwards from 27: ? Cube root: 3 Raise to the power of 4: 81 What power should I raise both sides of 3 the equation to ‘cancel’ the 4 power? 𝟒 𝟑 𝟑 𝒙𝟒 = 𝟒 ? 𝒙 = 𝟐𝟕𝟑 𝒙 = 𝟖𝟏 𝟒 𝟐𝟕𝟑 Further Examples Solve 𝑥 2 − 3 = 7 2 9 𝟐𝟓 𝒙 = 𝟗 𝟑 − ? 𝟐𝟓 𝟐 𝟐𝟕 𝒙= = 𝟗 𝟏𝟐𝟓 𝟐 −𝟑 Solve 𝑦 −3 = 3 3 8 𝟐𝟕 = 𝟖 ? −𝟏 𝟐𝟕 𝟑 𝟐 𝒚= = 𝟖 𝟑 𝒚−𝟑 Test Your Understanding 2 3 Solve 𝑥 = 9 𝑥= Solve 𝑥 3 −2 = 𝟑 𝟗𝟐? = 𝟐𝟕 8 27 𝟖? 𝑥= 𝟐𝟕 𝟐 −𝟑 𝟗 = 𝟒 Exercise 4 2 3 1 If 𝑥 = 9, find 𝑥. Questions on provided worksheet. 𝒙 =?𝟐𝟕 4 3 𝒚 =?𝟖 2 Solve 𝑦 = 16 3 2 3 [AQA FM June 2012 Paper 1] 𝑥 = 8 and 25 𝑦 −2 = 4 . 𝑥 Work out the value of 𝑦 7 𝒙 ÷ 𝒚 = 𝟒?÷ 𝟓 = 𝟏𝟎 4 [AQA FM2June 2013 Paper 1] 1 Solve 𝑥 −3 = 7 9 writing your answer as a 𝟐 proper fraction. 𝟐𝟕 𝟓𝟏𝟐 ? 𝑝 = 𝒒𝟔 × [AQA FM Set 1 Paper 2] You are given that 𝑥 = 5𝑚 and 𝑦 = 5𝑛 . (a) Write 5𝑚+2 in terms of 𝑥. 𝟐𝟓𝒙 ? 𝑚−𝑛 (b) Write 5 in terms of 𝑥 and 𝑦. 𝒙 𝒚 (c) Write −2 6 4 5 [June 2013 Paper 2] 𝑝 = 𝑞 × 𝑟 Write 𝑝 in terms of 𝑞 and 𝑟. Give your answer in its simplest form. 𝟏 − 𝒓𝟒 𝟐 FM Set 3 Paper 1] 6 [AQA 1 𝑥 2 = 6 and 𝑦 −3 = 64 𝑥 Work out the value of 𝑦 𝟏 𝒙 ÷ 𝒚 = 𝟑𝟔?÷ = 𝟏𝟒𝟒 𝟒 = ? 𝒒−𝟑 × 𝒓−𝟐 53𝑛 (d) Write 5 ? in terms of 𝑦. 𝒙𝟑 ? 𝑚+𝑛 2 in terms of 𝑥 and 𝑦. 𝒙𝒚 𝒐𝒓 ? 𝒙 𝒚 Changing bases What do you notice about all of the numbers: 2, 8, 4 They’re all powers of 2! We could replace the numbers with 21 , 23 and 22 so that we have a?consistent base. Changing bases 1 𝑥 10 Solve 4 = 2 2 Solve 2𝑥 = 83 2 First convert everything to powers of 2. 22 𝑥 ?= 10 2 ? 210 22𝑥 = 𝑥= ? 5 3 𝟐𝒙 = 𝟐 𝟑 𝟑 𝟏 𝟐𝟐 𝟐 𝒙 = 𝟐𝟗 ÷ 𝒙 = 𝟖. 𝟓 First convert everything to powers of 2… 𝟏 𝟐𝟐 ? = 𝟐𝟖.𝟓 If 2 2 = 2𝑘 , determine 𝑘. 𝟐 𝟐= 𝟑 𝒌= 𝟐 𝟐𝟏 × ? 𝟏 𝟐𝟐 = 𝟑 𝟐𝟐 Test Your Understanding 1 If 9 3 = 3𝑘 , find 𝑘. 𝟐 𝟗 𝟑=𝟑 × 𝟓 ? 𝒌= 𝟐 2 𝟏 𝟑𝟐 = Solve 3𝑥 = 92𝑥−1 𝒙 𝟐 𝟐𝒙−𝟏 𝟑 = 𝟑 𝟑𝒙 = 𝟑𝟒𝒙−𝟐 𝒙 = 𝟒𝒙 − 𝟐 ? 𝟐 = 𝟑𝒙 𝟐 𝒙= 𝟑 𝟓 𝟑𝟐 Exercise 5 1 Write as a single power of 2: a 410 = 𝟐 ?= 𝟐𝟏𝟎 b 8𝑥 × 24 = 𝟐𝟑𝒙+𝟒 ? 𝟐 𝟏𝟎 𝟒 2 3 f 4 3 2 1 = 𝟐𝟐 𝟏 𝟐𝟑 1 5 ? =𝟐 𝟑 5 𝟏𝟗 g 43 × 8 = 𝟐𝟑 × 𝟐𝟓?= 𝟐𝟏𝟓 [Edexcel GCSE(9-1) Nov 2017 2F Q21c, Nov 2017 2H Q6c Edited] 100𝑎 × 1000𝑏 can be written in the form 10𝑤 Express 𝑤 in terms of 𝑎 and 𝑏. 𝒘 = 𝟐𝒂 + 𝟑𝒃 3 Solve for 𝑥: 𝒙=𝟑 a 8 𝑥 = 29 𝟑 b 5𝑥 = 5 5 𝒙= ? 86 d 27𝑥 = 930 e 42𝑥+1 = 82𝑥−1 𝑥 3 84 [Edexcel IGCSE Jan2017-4H Q16d] 1 𝑛 3 4 = 3 9 Work out the exact value of 𝑛. 𝟒 𝒏=− 𝟑 ? 2 c 4𝑥 = 24 1 165 ? 𝟓 𝟑 𝟐 [Edexcel GCSE(9-1) June 2017 2H Q18] ×2 = Work out the exact value of 𝑥. 𝒙 = 𝟏. 𝟒𝟓 𝟏𝟔 𝟑 c 16 = ?= 𝟐 d 2𝑥 × 4𝑦 × 8𝑧 × 16 = 𝟐𝒙+𝟐𝒚+𝟑𝒛+𝟒 ? 𝟏 𝟑 𝟐 = 𝟐𝟐 e 2 2 = 𝟐𝟏 × 𝟐? 𝟐𝟒 𝟑 4 ? ?𝟐 𝒙 =?𝟕 𝒙 =?𝟏𝟎 𝒙= ? 𝟓𝟐 N Solve 3 4 9 × 27 = 𝑥 3 𝟏 𝟐 𝟑 𝟑 × 𝟏 𝟑 𝟒 𝟑 𝟐 𝟑 𝟑𝟑 × 𝟑𝟒 = 𝟏 𝟏𝟕 𝟑𝟏𝟐 = 𝟑𝒙 𝟏𝟕 𝟏 = 𝟏𝟐 𝒙 𝟏𝟐 𝒙= 𝟏𝟕 𝟏 𝟑𝒙 ? = 𝟏 𝟑𝒙
