Formula sheet
Law of indices (quotient)
Given on the exam formula sheet, you are not expected to recall it. You still need to recognise a division of powers with the same base, and to subtract the bottom index from the top index (top − bottom), leaving the base as it is.
What the symbols mean
- a the base, the number or letter being raised to a power
- m the index (power) of the top term (numerator)
- n the index (power) of the bottom term (denominator)
Given in the exam, or memorise?
Given on the exam formula sheet, you are not expected to recall it. You still need to recognise a division of powers with the same base, and to subtract the bottom index from the top index (top − bottom), leaving the base as it is.
Why it works
Dividing powers cancels equal factors on the top and bottom.
- am ÷ an is m factors of a over n factors of a.
- Each factor on the bottom cancels one on the top, removing n of them.
- What remains is (m − n) factors of a, that is am−n.
Worked example 1
Express 5⁷ ÷ 5⁴ as a single power of 5, then find its value.
- Same base 5, so apply am ÷ an = am−n.
- Subtract the indices (top − bottom): 5⁷ ÷ 5⁴ = 57−4 = 5³.
- Evaluate 5³ = 125.
Worked example 2
Simplify 6y² ÷ 2y⁵, giving your answer with a positive index.
- Divide the coefficients separately: 6 ÷ 2 = 3.
- Same base y, subtract the indices (top − bottom): y² ÷ y⁵ = y2−5 = y⁻³.
- Rewrite the negative index as a reciprocal: 3y⁻³ = 3/y³.
Where students go wrong
- Dividing or subtracting the bases too, the base stays the same, only the indices are subtracted.
- Subtracting the wrong way round, it is always the top index minus the bottom index, or the sign of the answer flips.
- Forgetting that a negative index means a reciprocal, y⁻³ = 1/y³, and that coefficients are divided separately from the powers.
Use it with
Source:SPM: Format Pentaksiran mulai 2021, Matematik (1449)
Frequently asked questions
Do I subtract the top or the bottom index first?
Always take the top (numerator) index minus the bottom (denominator) index: am ÷ an = am−n. So 5⁷ ÷ 5⁴ = 57−4 = 5³.
Keep the same base and subtract strictly in that order, otherwise you may end up with the sign of the index wrong.
What if the answer has a negative index?
A negative index is fine, it simply means a reciprocal. For example y² ÷ y⁵ = y2−5 = y⁻³, which equals 1/y³.
Subtract the indices as normal, then rewrite the negative power as one over the positive power whenever a positive form is asked for.
Does this rule change the coefficients too?
Yes, you divide them separately. For 12x⁸ ÷ 3x², divide the coefficients (12 ÷ 3 = 4) and subtract the indices (8 − 2 = 6), which gives 4x⁶.
The index law handles the powers, while ordinary division handles the numbers in front of the letters.