What are the basic laws of indices?

For the same base, add the indices when multiplying and subtract them when dividing; a power of a power multiplies the indices; any non-zero number to the power 0 is 1.

How should a fractional index be read?

The denominator is the root and the numerator is the power, so 8^(2/3) takes a cube root first and then squares it.

What does a negative index mean?

It means a reciprocal: a^-n = 1 / a^n, so 16^(-3/4) is 1/8.

How are logarithms and indices related?

log base a of N asks which power of a gives N, so the two statements say the same thing in different notation.

S4 Maths indices and logarithms paper 1: index laws and the log definition

2026-11-10

Tutor Prep Guide · S4 Maths Indices and Logarithms

Ten items start from the index laws and then bring in fractional indices, negative indices and the definition of a logarithm, each with an answer and a one-line note.

Direct answers　Items 01 to 03 cover multiplying powers, a power of a power and the zero index, 04 to 06 fractional and negative indices and a simple exponential equation, 07 to 09 scientific notation and log values, and 10 a wrong product of indices.

01　Paper details (check answers afterwards)

Item	Detail
Questions	10
Suggested time	25
Difficulty mix	Foundation 3 · Intermediate 3 · Challenge 3 · Error redo 1
Chapter	Chapter: S4 Maths Indices and Logarithms
Level	S4

02　Questions (10 items)

FoundationWork out 2^3 x 2^4 as a whole number.
FoundationWork out (3^2)^3.
FoundationWork out 5^0 + 5^-1 as a decimal.
IntermediateWork out 8^(2/3).
IntermediateWork out 16^(-3/4) as a fraction.
IntermediateSolve 2^x = 32.
ChallengeWrite 0.00042 in scientific notation.
ChallengeFind the value of log base 2 of 64.
ChallengeFind log base 3 of 81 plus log base 3 of 1/9.
Error redoA student writes 2^3 x 2^4 = 2^12. What went wrong and what is the correct value?

03　Answers and one-line notes

#	Answer	One-line note
1	128	For the same base, add the indices: 3 + 4 = 7, so 2^7 = 128.
2	729	A power of a power multiplies the indices: 2 x 3 = 6, giving 3^6 = 729.
3	1.2	Any non-zero number to the power 0 is 1, and 5^-1 = 1/5 = 0.2, so the total is 1.2.
4	4	The denominator 3 means a cube root: 8^(1/3) = 2, then square it to get 4.
5	1/8	A negative index means a reciprocal: 16^(3/4) = 8, so the answer is 1/8.
6	x = 5	Write 32 as a power of 2: 32 = 2^5, so comparing indices gives x = 5.
7	4.2 x 10^-4	The point moves four places to the right, so the index is -4 and the form is 4.2 x 10^-4.
8	6	Which power of 2 gives 64? Since 2^6 = 64, the logarithm is 6.
9	2	log base 3 of 81 is 4 and log base 3 of 1/9 is -2, so the sum is 2.
10	The indices must be added, so the answer is 2^7 = 128	Multiplying powers of the same base adds the indices rather than multiplying them: 3 + 4 = 7, so 2^7 = 128.

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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Questions and answers are written originally by the TutorZone team and are not taken from any school, publisher or examination authority paper. Every value is checked by computer.