Index Laws
Q1. Simplify using the multiplication law (am × an = am+n).
(a) 23 × 24 (b) 52 × 55 (c) x4 × x3
(d) 106 × 102 (e) a7 × a (f) 35 × 30
Q2. Simplify using the division law (am ÷ an = am−n).
(a) 26 ÷ 22 (b) 108 ÷ 103 (c) x9 ÷ x4
(d) 75 ÷ 75 (e) a3 ÷ a (f) 54 ÷ 50
Q3. Simplify using the power of a power law ((am)n = amn).
(a) (23)2 (b) (52)3 (c) (104)2
(d) (x3)4 (e) (a2)5 (f) (30)10
Q4. Simplify fully.
(a) 23 × 24 ÷ 22 (b) (32)3 ÷ 34 (c) x5 × x3 ÷ x4
(d) (a4)2 × a3 (e) 107 ÷ (102 × 103)
Q5. A scientist measures a bacteria population as 212 cells. After treatment, the population is reduced to one-quarter of its original size.
(a) Write the new population as a power of 2. 1 mark
(b) If the population then doubles, what is it now? Write as a single power of 2. 1 mark
Q6. Show that 82 × 43 can be written as a single power of 2. 2 marks
Q7. A student claims that am + an = am+n.
(a) Test this claim with a = 2, m = 2, n = 3. 1 mark
(b) Is the claim true? Explain. 2 marks
(c) Write a general rule for when am × an = am+n and explain why addition does not work the same way. 2 marks
Answer Key
Q1. (a) 27 (b) 57 (c) x7 (d) 108 (e) a8 (f) 35
Q2. (a) 24 (b) 105 (c) x5 (d) 70 = 1 (e) a2 (f) 54
Q3. (a) 26 (b) 56 (c) 108 (d) x12 (e) a10 (f) 30 = 1
Q4. (a) 25 (b) 32 (c) x4 (d) a11 (e) 102
Q5. (a) 212 ÷ 22 = 210 (b) 210 × 21 = 211
Q6. 82 = (23)2 = 26; 43 = (22)3 = 26; 26 × 26 = 212
Q7. (a) 22 + 23 = 4 + 8 = 12, but 25 = 32. Not equal. (b) False. Addition of powers does not combine the indices. (c) Multiplication law: am × an = am+n because m factors of a multiplied by n factors of a gives m+n factors. Addition cannot be rewritten as repeated multiplication of the same base.
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