Working with Exponents | Moosa Academy

When two powers share the same base, four rules cover everything you can do with them. Three of those rules are quick shortcuts — and the fourth is really a warning, because addition and subtraction do not work the way the others do.

Rule 1 Multiplying — add the exponents

a^m \times a^n = a^{\,m+n}

3^2 \times 3^2 = 3^{2+2} = 3^4 = 81

This makes sense once you write it out: two copies of 3 multiplied by two more copies gives four copies in total.

Rule 2 Dividing — subtract the exponents

a^m \div a^n = a^{\,m-n}

3^2 \div 3^2 = 3^{2-2} = 3^0 = 1

This is also where $a^0 = 1$ comes from. Any quantity divided by itself is 1, and the rule turns that into an exponent of zero.

Rule 3 Adding or subtracting — factor instead

There is no rule for adding or subtracting exponents directly. Instead, take out the common factor.

3^2 + 3^2 = 3^2(1+1) = 3^2 \times 2 = 9 \times 2 = 18

3^2 - 3^2 = 3^2(1-1) = 3^2 \times 0 = 0

Note what did not happen: $3^2 + 3^2$ is not $3^4$ . Adding powers leaves the exponent alone and changes the coefficient instead.

Rule 4 A power raised to a power — multiply

\left(a^m\right)^n = a^{\,m \times n}

\left(3^2\right)^2 = 3^{2 \times 2} = 3^4 = 81

Compare this with Rule 1 carefully. Multiplying two powers adds the exponents; raising a power to a power multiplies them. Mixing these two up is the most frequent error in the topic.

Note The four rules together

Every rule here requires the same base. With different bases none of them apply.

Example Simplifying a fraction with a variable exponent

Simplify $\dfrac{2^{\,x+1}}{2^{\,x}}$ .

The bases match, so the division rule applies.

Subtract the exponents:

(x+1) - x = 1

= 2^1 = 2

⟹ 2

The same answer comes from splitting the numerator instead:

2^{\,x+1} = 2^{\,x} \times 2

by the multiplication rule.

So the fraction becomes

\dfrac{2^{\,x} \times 2}{2^{\,x}}

, and

2^{\,x}

cancels.

⟹ 2, as before

Example Choosing the right rule

Evaluate $2^3 \times 2^4$ , $\left(2^3\right)^4$ and $2^3 + 2^3$ .

Multiplication adds:

2^{3+4} = 2^7 = 128

Power of a power multiplies:

2^{3 \times 4} = 2^{12} = 4096

Addition factors:

2^3(1+1) = 8 \times 2 = 16

⟹ 128, 4096 and 16

Three expressions built from the same numbers give three completely different answers. Identifying the operation before touching the exponents is what keeps them apart.

Note Mistakes to avoid

Writing

3^2 + 3^2 = 3^4

— addition does not combine exponents at all.

Adding exponents when the expression is a power raised to a power.

Multiplying exponents when two powers are simply being multiplied.

Applying any of these rules when the bases are different.

Forgetting that

a^0 = 1

, which follows directly from the division rule.

Subtracting in the wrong order — it is always the numerator's exponent minus the denominator's.

Summary

All four rules require the same base.
Multiplying powers adds the exponents: aᵐ × aⁿ = aᵐ⁺ⁿ.
Dividing powers subtracts them: aᵐ ÷ aⁿ = aᵐ⁻ⁿ, which is also why a⁰ = 1.
Adding or subtracting powers has no exponent rule — take out the common factor instead.
A power raised to a power multiplies the exponents: (aᵐ)ⁿ = aᵐⁿ.