Formula sheet

Law of indices (power)

Q: Why do I multiply the indices instead of adding them?

Because (am)n means am written n times and all multiplied together. Each copy has m factors of a, and there are n copies, so the total is m × n factors: (am)n = amn. Adding indices is only for am × an, where you multiply two separate powers.

Q: What is the difference between (a m ) n and a m × a n ?

In (am)n you raise one power to another power, so you multiply the indices: amn. In am × an you multiply two powers, so you add the indices: am+n. The base stays the same in both cases, it is only the operation on the indices that differs.

This is one of the index laws given in the exam. You do not memorise it, but you must spot when a power is being raised to another power, and remember to multiply the two indices, not add them, while keeping the same base.

Written by the spmmath.com.my editorial team.

Law of indices (power)

(a^m)ⁿ = a^mn

Given in the exam

What the symbols mean

a the base, the number or letter being raised to a power
m the inner index, the power inside the bracket
n the outer index, the power the whole bracket is raised to

Given in the exam, or memorise?

Why it works

Raising a power to a power stacks the same factors again and again.

(a^m)ⁿ means the block a^m is multiplied by itself n times.
Each block contributes m factors of a, and there are n blocks.
Altogether that is m × n factors, giving a^mn.

Worked example 1

Express (2⁴)³ as a single power of 2, then find its value.

A power is raised to another power, so apply (a^m)ⁿ = a^mn.
Multiply the indices: (2⁴)³ = 2^4×3 = 2¹².
Evaluate 2¹² = 4096.

Worked example 2

Simplify (x⁵)⁴.

Apply (a^m)ⁿ = a^mn, multiply the two indices, do not add them.
Multiply: (x⁵)⁴ = x^5×4 = x²⁰.
So the simplified form is x²⁰ (adding by mistake would wrongly give x⁹).

Where students go wrong

Adding the indices instead of multiplying, (a⁴)⁵ is a²⁰, not a⁹.
Confusing this rule with a^m × aⁿ, which adds the indices; a power-of-a-power multiplies them.
When a coefficient sits inside the bracket, forgetting to raise it too, in (3x²)³ the 3 becomes 3³ = 27, giving 27x⁶.

Use it with

Source:SPM: Format Pentaksiran mulai 2021, Matematik (1449)

Book a Trial ClassOne-hour paid trial · Same-day reply · from RM50/hr

Frequently asked questions

Why do I multiply the indices instead of adding them?

Because (a^m)ⁿ means a^m written n times and all multiplied together. Each copy has m factors of a, and there are n copies, so the total is m × n factors: (a^m)ⁿ = a^mn.

Adding indices is only for a^m × aⁿ, where you multiply two separate powers.

What is the difference between (a^m)ⁿ and a^m × aⁿ?

In (a^m)ⁿ you raise one power to another power, so you multiply the indices: a^mn. In a^m × aⁿ you multiply two powers, so you add the indices: a^m+n.

The base stays the same in both cases, it is only the operation on the indices that differs.

Do I raise the number in front as well, like in (3x²)³?

Yes, everything inside the bracket is raised. So (3x²)³ means 3³ × (x²)³ = 27 × x⁶ = 27x⁶.

Many students raise only the letter and forget the coefficient, leaving 3x⁶. Apply the outer power to the number and the variable together to stay correct.