An exponent is a short way to write repeated multiplication: 2³ means 2 × 2 × 2, and the exponent tells you how many copies of the base to multiply. Everything else about powers, including negative and fractional exponents, follows from that single idea, so one good mental model covers the whole topic.

Key Takeaways

2³ = 8, 5⁰ = 1, and 2⁻² = 0.25: the base pattern is repeated multiplication.

Multiply powers with the same base by adding exponents; divide by subtracting them.

A fractional exponent like 27^(1/3) is the same as a cube root (∛27 = 3).

On the calculator, type powers with the ^ key, for example 2^10.

What is an exponent?

An exponent, also called a power, is the small number that says how many times to multiply the base by itself. In 2³ the base is 2 and the exponent is 3, meaning 2 × 2 × 2 = 8. The exponent appears in the text as a superscript, but on a calculator you type it with the ^ key: 2^3.

Worked example 1: repeated multiplication. 5² = 5 × 5 = 25. 3⁴ = 3 × 3 × 3 × 3 = 81. Type 5^2 and 3^4 to confirm both on any scientific calculator.

The rules for multiplying and dividing powers

When powers share the same base, the arithmetic collapses to a single exponent. These three rules are the workhorses, and the log and exponent rules guide shows them side by side with their logarithm cousins.

Product rule: add the exponents

When you multiply two powers of the same base, add the exponents: aⁿ × aᵐ = aⁿ⁺ᵐ.

Worked example 2: product rule. 2³ × 2⁴ = 2⁷. Check the two sides by hand: 2³ = 8, 2⁴ = 16, and 8 × 16 = 128; likewise 2⁷ = 128. Both paths give the same number, which is the whole point of the rule.

Quotient rule: subtract the exponents

When you divide powers of the same base, subtract the exponents: aⁿ ÷ aᵐ = aⁿ⁻ᵐ.

Worked example 3: quotient rule. 2⁵ ÷ 2² = 2³. On the left, 32 ÷ 4 = 8; on the right, 2³ = 8. The shortcut works because the two 2 × 2 factors in the denominator cancel two of the five factors upstairs.

Power of a power: multiply the exponents

Raise a power to another power and the exponents multiply: (aⁿ)ᵐ = aⁿᵐ.

Worked example 4: power of a power. (2³)² = 2⁶. Writing it out, 2³ = 8 and 8² = 64; directly, 2⁶ = 64. On the calculator, be sure to type the parentheses: (2^3)^2 = 64. Without them, 2^3^2 is read as 2^(3²) = 2⁹ = 512, because powers bind to the right.

Zero and negative exponents

The rules above decide the weird-looking cases automatically. Zero as an exponent always gives 1 for any nonzero base, because aⁿ ÷ aⁿ = 1 and the quotient rule says the answer is a⁰. So 5⁰ = 1, 2⁰ = 1, and on the calculator 10^0 → = 1.

A negative exponent means a reciprocal: a⁻ⁿ = 1 ÷ aⁿ. It never makes the answer negative, only smaller.

Worked example 5: negative exponents. 2⁻² = 1 ÷ 2² = 1 ÷ 4 = 0.25. 2⁻¹ = 1 ÷ 2 = 0.5. 10⁻⁶ = 1 ÷ 10⁶ = 0.000001. Type 2^-2 and 10^-6 to match these.

Fractional exponents and roots

A fractional exponent is a root in disguise. a^(1/2) is the square root, a^(1/3) is the cube root, and a^(1/n) is the nth root. The denominator of the fraction names the root, and the numerator, when larger than 1, adds an extra power.

Worked example 6: fractional exponents. 9^(1/2) = √9 = 3. 8^(1/3) = ∛8 = 2. 16^(1/4) = 2, since 2⁴ = 16. Combining the ideas, 27^(2/3) means the cube root of 27 squared: 3² = 9. This equivalence runs both directions, and the square and cube roots post leans on it heavily.

Notation, order of operations, and calculator tips

On the calculator, keep these three habits in mind:

Type powers with the ^ key: 2^10 → = 1024.
Wrap the base in parentheses when it is negative or an expression: (-5)^2 = 25, not -25, because -5^2 is read as -(5²).
Watch out for stacked powers: ^ binds to the right, so 2^3^2 = 2⁹ = 512. Use parentheses when you mean (2³)² = 64.

Powers sit above multiplication and division and below parentheses in the order of operations, which is why 2 × 3² is 18 and not 36. When exponents get huge, the result is often written in scientific notation: 10^6 = 1,000,000 is a million, and 2^20 = 1,048,576. The scientific notation calculator turns those long digit strings back into a compact power of ten.

Why exponents matter beyond the classroom

Powers name the numbers that recur in science and engineering: doubling is multiplication by powers of 2, compound growth multiplies by (1 + r)^t year after year, and logarithmic scales compress powers of ten into readable graphs. Understanding the six rules here, product, quotient, power-of-power, zero, negative, and fractional, means you can convert between any of these forms on the fly, and the exponent calculator does the arithmetic for you when the numbers stop being tidy.

Frequently asked questions

Why does any nonzero number to the power of zero equal 1?

Because aⁿ ÷ aⁿ = 1, and the quotient rule turns that same expression into a⁰. The definition is chosen so the exponent rules stay consistent, not because something is being multiplied zero times.

What does a negative exponent mean?

It means the reciprocal, not a negative result. 2⁻² = 1 ÷ 2² = 0.25. You can read a⁻ⁿ as “one over a to the n”.

How do I type a power on the calculator?

Use the ^ key. 2^10 → = 1024. For a negative base, wrap it in parentheses: (-5)^2 → = 25.

What is the difference between `2^3^2` and `(2^3)^2`?

With no parentheses, powers bind to the right, so 2^3^2 = 2⁹ = 512. The parenthesized form (2^3)^2 = 8² = 64. Five hundred twelve and sixty-four are very different answers, so the parentheses matter.