What are the four basic index laws?

Add indices when multiplying like bases, subtract when dividing, multiply for a power of a power, and any non-zero base to the power 0 is 1; a negative index gives a reciprocal.

Why must the base stay unchanged?

3 squared times 3 cubed is five 3s multiplied together, not five 9s; the base only becomes 9 if the product is raised as a whole.

S5 Maths Indices and Logarithms Paper 1: Basics

2026-11-09

Tutor Prep Guide · S5 Maths

Ten original index items, each with an answer and a one-line explanation.

Direct answers　Items 1-3 cover multiplying, dividing and powering indices, 4-6 handle zero and negative indices, 7-9 move between index and logarithm form, and item 10 unpacks the common base error.

01　Paper details (check answers afterwards)

Item	Detail
Questions	10
Suggested time	20
Difficulty mix	Foundation 3 · Intermediate 3 · Challenge 3 · Error redo 1
Chapter	Chapter: S5 Maths: Indices and Logarithms
Level	S5

02　Questions (10 items)

FoundationWork out 2 cubed times 2 to the fourth.
FoundationWork out 5 to the 6th divided by 5 squared.
FoundationSimplify (3 squared) cubed.
IntermediateFind the value of 7 to the power 0.
IntermediateFind the value of 2 to the power minus 3 as a fraction.
IntermediateSimplify (2 x cubed) squared.
ChallengeSolve 2 to the x equals 32.
ChallengeFind the value of log base 2 of 8.
ChallengeFind the value of log base 10 of 1000.
Error redoA pupil writes 3 squared times 3 cubed equals 9 to the 5th. What is wrong and what is the correct value?

03　Answers and one-line notes

#	Answer	One-line note
1	2 to the 7th, which is 128	For the same base, add the indices: 3 + 4 = 7, so the answer is 128; multiplying the indices gives 2 to the 12th, which is wrong.
2	5 to the 4th, which is 625	For the same base, subtract the indices: 6 – 2 = 4, and 5 to the 4th is 625.
3	3 to the 6th, which is 729	A power of a power multiplies the indices: 2 x 3 = 6, so the value is 729; adding them to get 3 to the 5th is a common slip.
4	1	Any non-zero number to the power 0 equals 1, not 0 and not 7.
5	1/8, that is 0.125	A negative index means a reciprocal: 1 divided by 2 cubed gives 1/8, not minus 8.
6	4 x to the 6th	Square both parts: 2 squared is 4 and the index of x becomes 3 x 2 = 6.
7	x = 5	32 is 2 to the 5th, and equal powers of the same base need equal indices, so x is 5.
8	3	A logarithm asks what power of 2 gives 8, and 2 cubed is 8, so the answer is 3.
9	3	10 cubed is 1000, so the logarithm is 3; a third of 1000 misunderstands the definition.
10	3 to the 5th, which is 243	The base stays the same and only the indices are added, giving 243; 9 to the 5th is 59 049, which is 243 times too big.

04　What to do after marking

After marking, redo only the wrong items and write down which step went wrong (formula, unit, or misread number).
Repeat the same item two days later; move to the next paper only when both attempts are correct.
Parents only need the answer and note columns; there is no need to explain every item.

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