Exponents and the exponent rules, explained

Updated October 20, 2026

An exponent tells you how many times to multiply a base by itself: 2⁵ means 2 × 2 × 2 × 2 × 2, which is 32. The exponent rules follow from that meaning: multiply powers of the same base by adding exponents, divide by subtracting them, and raise a power to a power by multiplying them. Any nonzero number to the power 0 is 1, and a negative exponent means a reciprocal.

What an exponent means

In 5³, the 5 is the base and the small raised 3 is the exponent, also called the power. It means three 5s multiplied together: 5 × 5 × 5 = 125. Read it as five cubed or five to the third power. A power of 2 is read as squared, because 4² is the area of a square with side 4.

Exponents are shorthand for repeated multiplication, just as multiplication is shorthand for repeated addition. They appear in area and volume, compound interest, population growth, computer memory (powers of 2), and very large or very small measurements written in scientific notation.

The most common early mistake is multiplying the base by the exponent. 2³ is 2 × 2 × 2 = 8, not 2 × 3 = 6.

The exponent rules

Each rule below is a shortcut you can rebuild by writing the multiplication out. For a nonzero base a and exponents m and n:

Product rule: aᵐ × aⁿ = aᵐ⁺ⁿ. For example x² × x³ = x⁵, because two x's times three x's is five x's.
Quotient rule: aᵐ ÷ aⁿ = aᵐ⁻ⁿ. For example 7⁶ ÷ 7⁴ = 7², because four 7s cancel.
Power of a power: (aᵐ)ⁿ = aᵐⁿ. For example (x³)² = x³ × x³ = x⁶.
Power of a product: (ab)ⁿ = aⁿbⁿ. For example (3x)² = 9x².
Power of a quotient: (a/b)ⁿ = aⁿ/bⁿ. For example (2/5)² = 4/25.
Zero exponent: a⁰ = 1.
Negative exponent: a⁻ⁿ = 1/aⁿ. For example 2⁻³ = 1/8.

Why the zero and negative exponent rules make sense

Look at the powers of 2 going down: 2⁴ = 16, 2³ = 8, 2² = 4, 2¹ = 2. Each step down divides by 2. Keep going: 2⁰ = 1, 2⁻¹ = 1/2, 2⁻² = 1/4. The pattern forces those values.

The quotient rule agrees. 5³ ÷ 5³ is 1, since any nonzero number divided by itself is 1, and by the rule it is also 5⁰. Similarly 5² ÷ 5⁵ is 1/5³, and by the rule it is 5⁻³.

So a negative exponent never makes a number negative. 2⁻³ is 1/8, a small positive number. It means a reciprocal.

Worked example: simplifying with several rules

Apply one rule per line, and handle the numbers and each letter separately.

ProblemSimplify $\frac{{(2 x^{3})}^{2} \cdot x^{- 4}}{8 x}$

${(2 x^{3})}^{2}$ $= 2^{2} \cdot x^{6}$ $= 4 x^{6}$
Power of a product: square the 2 and the x³. Power of a power: 3 × 2 is 6.
$4 x^{6} \cdot x^{- 4}$ $= 4 x^{2}$
Product rule: add the exponents, 6 + (−4) = 2.
$\frac{4 x^{2}}{8 x}$ $= \frac{4}{8} \cdot x^{2 - 1}$
Divide the numbers, and use the quotient rule on x. A plain x has exponent 1.
$\frac{1}{2} x$ $= \frac{x}{2}$
4/8 simplifies to 1/2, and x¹ is x.

Answer $\frac{x}{2}$

Scientific notation

Scientific notation writes a number as a value from 1 up to (but not including) 10, times a power of 10. The distance from Earth to the Sun, about 150,000,000 km, is 1.5 × 10⁸ km. The width of a typical human hair, roughly 0.00008 m, is 8 × 10⁻⁵ m. A positive power of 10 moves the decimal point right; a negative power moves it left.

To multiply numbers in scientific notation, multiply the front numbers and add the powers of 10, then adjust so the front number is between 1 and 10.

ProblemMultiply $(3 \times 10^{4}) (5 \times 10^{- 7})$

$(3 \times 5) \times (10^{4} \times 10^{- 7})$
Regroup: multiplication can be done in any order.
$15 \times 10^{- 3}$
3 × 5 is 15, and 4 + (−7) is −3.
$1.5 \times 10^{1} \times 10^{- 3}$
15 is not between 1 and 10, so write it as 1.5 × 10¹.
$1.5 \times 10^{- 2}$
Add the exponents again: 1 + (−3) is −2. As a decimal this is 0.015.

Answer $1.5 \times 10^{- 2}$

Fractional exponents and roots

In high school, exponents can be fractions. The denominator is a root and the numerator is a power: a^(1/2) is the square root of a, a^(1/3) is the cube root, and a^(m/n) is the nth root of a, raised to the power m. These meanings are chosen so that the product rule still works: 9^(1/2) × 9^(1/2) = 9¹ = 9, so 9^(1/2) must be 3.

Taking the root first usually keeps the numbers small.

ProblemEvaluate $8^{2 / 3}$

$8^{2 / 3}$ $= (^{3} \sqrt 8)^{2}$
The denominator 3 means cube root; the numerator 2 means square.
$^{3} \sqrt 8$ $= 2$
2 × 2 × 2 is 8.
$2^{2}$ $= 4$
Square the result. Check: 4 × 4 × 4 is 64, and 8² is 64 too.

Answer $4$

Common exponent mistakes

These errors show up in every grade from 6 to 12:

Multiplying base and exponent: 4³ is 64, not 12.
Multiplying exponents in a product: x² × x³ is x⁵, not x⁶. Exponents multiply only for a power of a power.
Squaring a sum term by term: (a + b)² is a² + 2ab + b², not a² + b². Try a = 3 and b = 4: 49, not 25.
Applying the exponent to the sign by mistake: −5² is −25, while (−5)² is 25.
Reading a negative exponent as a negative number: 10⁻² is 0.01, not −100.
Forgetting the coefficient in a power of a product: (3x)² is 9x², not 3x².

How to practise and check your work

When a rule feels shaky, write the power out in full. Seeing x × x × x × x × x makes the product rule obvious, and it is the fastest way to settle an argument with yourself.

To check a simplification, pick a small number such as x = 2 and evaluate both the original and your answer. If they agree, you are very likely right; if they disagree, look for an added exponent that should have been multiplied, or the other way round. Avoid x = 0 and x = 1, which hide mistakes.

Questions

Why is any number to the power of zero equal to 1?

Dividing a power by itself gives 1, and the quotient rule says aⁿ ÷ aⁿ = a⁰. The pattern of dividing by the base each time the exponent drops by 1 leads to the same value.

What does a negative exponent mean?

It means a reciprocal. x⁻ⁿ = 1/xⁿ, so 3⁻² = 1/9. The result is positive when the base is positive.

When do you add exponents and when do you multiply them?

Add them when multiplying powers with the same base, as in x² × x³ = x⁵. Multiply them when raising a power to a power, as in (x²)³ = x⁶.

What is 0 to the power of 0?

In school algebra 0⁰ is usually left undefined, because the zero exponent rule and the rule that 0 to any positive power is 0 point to different values. Some areas of higher math define it as 1 for convenience.

What grade learns exponents?

Whole number exponents appear in grade 6, the integer exponent rules and scientific notation in grade 8, and fractional exponents in Algebra I and Algebra II.