Expressions & Equations

Exponents

Writing and evaluating numerical expressions with whole-number exponents.

Intermediate20 min lesson3 min readUpdated August 12, 2026Author not yet attributed

A Shorthand for Repeated Multiplication

A single sheet of paper is folded in half, then in half again, then again — 10 times in total. Before reading on, predict how many layers thick the folded stack would be. The answer involves multiplying 2 by itself over and over, which raises a question: is there a faster way to write '2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2' than spelling out all ten multiplications?

There is, and it mirrors a shortcut you've already used before. Multiplication itself is shorthand for repeated addition — 4 × 3 means '3, four times.' An exponent takes that same idea one level further: it's shorthand for repeated multiplication.

Definition — Exponent

A small raised number showing how many times a base number is multiplied by itself. In 2⁵, the base is 2 and the exponent is 5, meaning 2 × 2 × 2 × 2 × 2.

Worked Example — Evaluating an Exponent

Evaluate 2¹⁰, the paper-folding problem from the opening. This means 2 multiplied by itself 10 times: 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 1,024. Ten folds would create 1,024 layers.

Worked Example — Distinguishing an Exponent from Multiplication

Evaluate 3⁴ and compare it to 3 × 4. 3⁴ means 3 × 3 × 3 × 3 = 81. 3 × 4 is a completely different calculation, equal to 12. The small raised exponent and a multiplication symbol represent two different operations, even when they involve the same two numbers.

2^{5} = 2 \times 2 \times 2 \times 2 \times 2 = 32

Tip

For a quick sanity check, remember that any number raised to the power of 1 equals itself, and any number raised to the power of 2 is that number multiplied by itself once — useful reference points before tackling a larger exponent.

Common Mistakes

Confusing an exponent with a multiplier, such as evaluating 3⁴ as 3 × 4 = 12 instead of 3 × 3 × 3 × 3 = 81.
An exponent tells how many times the base multiplies itself — write out the repeated multiplication if the meaning isn't automatic yet.
Multiplying the base by the exponent one time too few or too many, such as evaluating 2⁴ as 2 × 2 × 2 (three 2s) instead of four.
The exponent equals the exact count of times the base appears in the multiplication — for 2⁴, write out four 2s: 2 × 2 × 2 × 2.

Key Takeaways

An exponent is shorthand for repeated multiplication, the same way multiplication is shorthand for repeated addition.
In an expression like 2⁵, the base is the number being multiplied, and the exponent counts how many times it's multiplied by itself.
An exponent and a multiplication symbol represent different operations, even applied to the same two numbers.

Summary

Exponents give repeated multiplication its own fast notation. The next lesson uses variables — letters standing in for unknown numbers — to write algebraic expressions from real descriptions.

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