aᵐ × aⁿ = aᵐ⁺ⁿ. So 2³ × 2⁴ = 2⁷ = 128.

Simplify 2³ × 2⁴ — Laws of Exponents

Question

Simplify: $2^3 \times 2^4$

Solution — Step by Step

When we multiply two powers with the same base, we add the exponents. This is the law: $a^m \times a^n = a^{m+n}$ .

The base here is 2 in both terms, so the law applies directly.

2^3 \times 2^4 = 2^{3+4} = 2^7

We keep the base as 2 and simply add 3 and 4.

2^7 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 128

So the final answer is $\mathbf{2^7 = 128}$ .

Why This Works

Think about what $2^3$ and $2^4$ actually mean. $2^3$ is three 2s multiplied together, and $2^4$ is four 2s multiplied together. When we write them side by side, we end up with $3 + 4 = 7$ twos multiplied together — which is exactly $2^7$ .

2^3 \times 2^4 = \underbrace{(2 \times 2 \times 2)}_{3 \text{ twos}} \times \underbrace{(2 \times 2 \times 2 \times 2)}_{4 \text{ twos}} = \underbrace{2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2}_{7 \text{ twos}}

This is why the law $a^m \times a^n = a^{m+n}$ works — it’s not magic, it’s just counting how many times you’re multiplying the base.

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

Valid only when the base is the same on both sides.

Alternative Method

We can verify by expanding completely, without using the law at all.

2^3 = 8 \qquad \text{and} \qquad 2^4 = 16

8 \times 16 = 128

Same answer. The exponent law gives us a shortcut — imagine doing this with $2^{10} \times 2^{15}$ . Expanding would be painful; adding exponents gives $2^{25}$ instantly.

For MCQs, use the exponent law directly — never expand unless the numbers are very small and you’re stuck. Adding exponents takes 2 seconds; expanding $2^{10}$ does not.

Common Mistake

Multiplying the exponents instead of adding them.

Students often write $2^3 \times 2^4 = 2^{12}$ , confusing this with the power of a power rule: $(a^m)^n = a^{m \times n}$ .

The rule $a^m \times a^n = a^{m+n}$ is for multiplication of two separate terms. The rule $(a^m)^n = a^{m \times n}$ is when a power is raised to another power. These are two different situations — keep them separate.

$2^3 \times 2^4 \neq 2^{12}$ — that would be $(2^3)^4$ .

Question

Solution — Step by Step

Why This Works

Alternative Method

Common Mistake

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