Formula sheet
Law of indices (product)
This law is printed on the exam formula sheet, so you do not memorise it. What you must know is when to apply it, only when the bases are the same and the powers are being multiplied, and to add (not multiply) the indices while keeping the base unchanged.
What the symbols mean
- a the base, the number or letter being raised to a power
- m the index (power) of the first term
- n the index (power) of the second term
Given in the exam, or memorise?
This law is printed on the exam formula sheet, so you do not memorise it. What you must know is when to apply it, only when the bases are the same and the powers are being multiplied, and to add (not multiply) the indices while keeping the base unchanged.
Why it works
Powers are just repeated multiplication, and that is why the indices add.
- am means a multiplied by itself m times; an means a multiplied n times.
- Writing them side by side, a is now multiplied a total of (m + n) times.
- Collecting them back into one power gives am+n.
Worked example 1
Express 3² × 3⁴ as a single power of 3, then find its value.
- The bases are the same (both 3), so apply am × an = am+n.
- Add the indices: 3² × 3⁴ = 32+4 = 3⁶.
- Evaluate 3⁶ = 729.
Worked example 2
Simplify 4a³ × 5a⁶.
- Multiply the coefficients separately: 4 × 5 = 20.
- The base a is the same, so add the indices: a³ × a⁶ = a3+6 = a⁹.
- Combine the results: 20 × a⁹ = 20a⁹.
Where students go wrong
- Multiplying the bases as well, 2³ × 2⁴ is not 4⁷; the base stays the same and only the indices are added.
- Trying to combine powers when the bases are different, 2³ × 3² cannot be joined into one power.
- Forgetting to multiply the coefficients separately, in 4a³ × 5a⁶ the numbers give 4 × 5 = 20, not 4 + 5.
Use it with
Source:SPM: Format Pentaksiran mulai 2021, Matematik (1449)
Frequently asked questions
Do I add or multiply the powers when multiplying terms?
When you multiply powers with the same base, you add the indices: am × an = am+n. You never multiply the indices for this rule, multiplying indices belongs to (am)n instead.
And the base itself stays unchanged; only the exponents are added together to make the single answer.
Can I use this rule if the bases are different?
No. The law am × an = am+n only works when the two bases are identical.
For 2³ × 3² the bases differ, so you cannot combine them into a single power, just work out each value separately (8 × 9 = 72) instead and multiply the results.
What happens to the number in front, like in 4a³ × 5a⁶?
Treat the coefficient and the power separately. Multiply the coefficients first (4 × 5 = 20), then add the indices of the same base (a³ × a⁶ = a⁹).
The final answer is 20a⁹. Coefficients are multiplied while indices are added, do not mix the two steps together.