TL;DR: To rewrite a power without exponents, write the base out as many times as the exponent says and multiply. So 2⁵ becomes 2 × 2 × 2 × 2 × 2. Three sub-cases matter: a power of one writes the base once, a power of zero becomes 1 (not 0, and not the base), and a negative exponent becomes a fraction with the positive power in the denominator — 2⁻³ is 1 ÷ (2 × 2 × 2).
The basic rule
An exponent is a count of how many copies of the base are multiplied together. That is the entire definition, and expanding it is just writing the count out.
Four examples covering the common shapes:
- 3⁴ = 3 × 3 × 3 × 3 = 81
- 7² = 7 × 7 = 49
- 10⁶ = 10 × 10 × 10 × 10 × 10 × 10 = 1,000,000
- x³ = x · x · x
Note the last one. When the base is a variable rather than a number, the expansion is the final answer — you cannot multiply it out, so “x·x·x” is what is being asked for.
That is the usual reason this question is asked: not to compute a value, but to see the structure. Expanding a power makes it obvious why the exponent rules work, which is exactly what algebra instruction is aiming at.
Power of one
Any base to the first power is the base itself. 5¹ = 5, x¹ = x.
This looks trivial and it is the rule that makes the multiplication rule consistent. If you did not treat a lone variable as having an exponent of one, you could not add exponents when multiplying like terms.
Writing the exponent explicitly, even when it is 1, is a useful habit in algebra. Most errors in simplifying expressions come from forgetting that an unmarked variable has a power of one.
Power of zero
Any non-zero base to the power of zero is 1. Not zero, not the base.
This is the rule students most often doubt, and the reason is worth understanding rather than memorising. Follow the pattern downward: 2⁴ = 16, 2³ = 8, 2² = 4, 2¹ = 2. Each step divides by 2. Continue the pattern and 2⁰ must be 2 ÷ 2 = 1. The pattern only stays consistent if the zero power is 1.
The rule holds for every base except zero. 0⁰ is undefined, because it sits at the intersection of two conflicting rules — anything to the zero is 1, and zero to anything is 0 — and neither can be applied without contradiction.
Negative exponents
A negative exponent does not make the answer negative. It moves the power to the other side of a fraction.
2⁻³ = 1 / 2³ = 1 / (2 × 2 × 2) = 1/8.
The same rule with a variable: x⁻² = 1 / (x · x).
Two consequences are worth stating because they come up constantly. A negative exponent applied to a fraction flips it into a positive power: (2/3)⁻² = (3/2)². And a negative sign in front of the expression is a completely separate thing from a negative exponent: −2³ is −8, while 2⁻³ is 1/8. Confusing those two is the single most common error in this topic.
Fractional exponents, briefly
A fractional exponent is a root. x^(1/2) is the square root of x, and x^(1/3) is the cube root.
So “rewriting without exponents” for a fractional power means writing a radical: 8^(1/3) = ∛8 = 2. The general form is x^(m/n) = the n-th root of x, raised to m.
This case is worth knowing because it is the only one where a power cannot be written as repeated multiplication at all. Writing 9^(1/2) as 9 × 9 would be wrong, and it is a common mistake when students apply the repeated-multiplication rule mechanically.
Common errors, and why they happen
Four mistakes account for most of the wrong answers in this topic, and each has a specific cause.
Multiplying the base by the exponent. 2⁵ is not 10. The exponent counts copies of the base, it does not tell you what to multiply the base by. Writing the expansion out makes the difference visible immediately, which is why teachers ask for it.
Applying a negative exponent as a negative result. 2⁻³ expands to 1/8, not −8. The minus in the exponent describes position in a fraction, not sign.
Treating a zero power as zero. 5⁰ is 1. The pattern argument in the section above is the reason, and it is worth being able to reproduce it rather than only remembering the answer.
Expanding a fractional exponent as repeated multiplication. 16^(1/4) is the fourth root, which is 2. Writing 16 four times and multiplying gives 65,536, which is 16⁴ — the reciprocal operation rather than the requested one.
Each of these is a rule applied to the wrong feature of the notation. Expanding by hand forces you to decide which feature applies before you compute, which is why the exercise exists at all.
Why expand at all
Three practical uses beyond classroom exercises.
Checking a simplification. If you expand both sides of a claim — that x² · x³ equals x⁵, for example — the identity becomes obvious: two copies times three copies is five copies.
Spotting order-of-operations errors. Expanded form shows exactly what is being multiplied, which makes it clear that −3² is −9 rather than 9. The exponent applies to the base, and the negative sign is outside it.
Estimating magnitude. 2¹⁰ expanded is 1,024, which is why computer memory sizes cluster around numbers like 1,024 and 4,096. Seeing the multiplication makes the sizes intuitive.
If you want to check an expansion quickly, particularly with larger exponents where manual multiplication gets long, a power of two calculator shows the value alongside the multiplication, which is useful for verifying your own working rather than replacing it.
The short version
Write the base as many times as the exponent, then multiply. Power of one is the base, power of zero is 1, negative exponents become fractions, and fractional exponents become roots. Do not confuse a negative exponent with a negative result.
