Integer Exponents and Exponent Rules

Start with the base

In 5³, the base is 5 and the exponent is 3. The expression means 5 × 5 × 5, so its value is 125. It does not mean 5 × 3. Before choosing an exponent rule, identify exactly what belongs to the base. Parentheses can make that base a negative number, a fraction, a product, or an entire expression.

For example, (−4)² means (−4)(−4), which equals 16. In −4², the exponent applies to 4 before the leading negative sign, so the result is −16. These expressions differ because the grouping differs. A calculator can reproduce either result depending on how the expression is entered; it cannot decide which expression the question intended.

Positive integer powers are easy to inspect by writing out the factors. That provides a useful check when a remembered rule feels uncertain. Expand a small example first, then return to the compact exponent notation.

Multiplying powers with the same base

The product rule is a^m × a^n = a^(m+n). For positive exponents, this simply counts all the repeated factors. In 2³ × 2⁴, the first factor contains three twos and the second contains four more. Together there are seven, giving 2⁷ = 128.

The bases must match for this form of the rule. In 2³ × 3⁴, adding the exponents would discard information about which number is being multiplied. Evaluate each power separately or use another justified rearrangement. There is no general rule that turns that expression into either 5⁷ or 6⁷.

Coefficients also remain separate. In (3x²)(4x⁵), multiply 3 by 4 and combine the powers of x: 12x⁷. The coefficient becomes 12 because of ordinary multiplication; the exponent becomes 7 because there are seven factors of x. Mixing these steps is a common source of answers such as 7x¹⁰.

Dividing powers with the same base

For a nonzero base, a^m ÷ a^n = a^(m−n). The subtraction follows from canceling matching factors. For example, 7⁵/7² leaves three factors of 7 after two factors cancel from the numerator and denominator. The result is 7³.

Order matters. The denominator exponent is subtracted from the numerator exponent. In x²/x⁶, the result is x^(2−6) = x^(−4), which is 1/x⁴. Reversing the subtraction changes the value. With x = 2, the original expression is 4/64 = 1/16, so an answer of 16 cannot be right.

Division also brings a restriction: the denominator cannot be zero. Even if x³/x simplifies to x², the original expression is undefined when x = 0. Keep that restriction if the question asks for an equivalent expression together with its domain.

A power raised to a power

The power rule is (a^m)^n = a^(mn). Here the exponents multiply. In (x³)⁴, each of the four repeated factors contributes three copies of x. The total is twelve copies, so the expression is x¹².

Compare that with x³ × x⁴ = x⁷. The first expression repeats the entire power; the second multiplies two separate powers. Their visual similarity can be misleading. Look at the parentheses and the operation between the terms before choosing addition or multiplication of exponents.

When the base is a product, the outer power applies to each factor: (ab)^n = a^n b^n. Thus (2x³)² = 2²(x³)² = 4x⁶. Leaving the coefficient as 2 means only part of the parenthesized expression was squared.

Zero powers and negative powers

For any nonzero number a, a⁰ = 1. One way to see the rule is to divide a power by itself. For a ≠ 0, a³/a³ equals 1, while the quotient rule gives a^(3−3) = a⁰. Both descriptions must agree. This reasoning does not establish a value for 0⁰ because the original division would be undefined.

A negative exponent indicates a reciprocal: a^(−n) = 1/a^n for a nonzero base. For example, 5^(−2) = 1/25. The negative sign in the exponent does not make the answer negative. It changes whether the factor appears in the numerator or denominator.

This distinction also explains why (−5)^(−2) is positive. First take the reciprocal of (−5)², then evaluate the square: 1/25. By contrast, (−5)^(−3) = −1/125 because an odd power of a negative base remains negative.

Fractions raised to a power

For a fraction with a nonzero denominator, (a/b)^n = a^n/b^n. Squaring 3/4 gives 9/16. A negative exponent first reverses the fraction: (3/4)^(−2) = (4/3)² = 16/9. Both numerator and denominator must be nonzero when the original fraction is raised to a negative power.

An efficient check is to think about size. The fraction 3/4 lies between zero and one, so its positive square is smaller than 3/4. Its reciprocal is greater than one, so the negative square should also be greater than one. This catches a reversal mistake without completing every calculation twice.

Do not distribute an exponent across addition. In general, (a + b)² is not a² + b². Using a = 2 and b = 3 shows the difference: the left side is 25 and the proposed right side is 13. Expanding the product produces a² + 2ab + b², including the middle term that the shortcut loses.

A combined worked example

Simplify (6x³y^(−2))/(2x^(−1)y), with x and y both nonzero. Start with the coefficients: 6/2 = 3. For x, subtract the denominator exponent from the numerator exponent: 3 − (−1) = 4. For y, the denominator exponent is an unwritten 1, so −2 − 1 = −3.

The result is 3x⁴y^(−3), or 3x⁴/y³ with positive exponents. The negative denominator exponent on x increased the resulting exponent because subtracting a negative adds. The positive denominator exponent on y made the resulting exponent more negative.

Check with simple values that satisfy the restrictions. If x = 1 and y = 2, the original expression becomes (6 × 1 × 1/4)/(2 × 1 × 2) = 3/8. The simplified expression gives 3 × 1/8 = 3/8 as well. A numerical check does not replace the algebraic argument, but it can expose a sign error quickly.

Choosing the rule from the expression

When separate powers with the same base are multiplied, add the exponents. When they are divided, subtract denominator from numerator. When a power is raised to another power, multiply the exponents. When a factor has a negative exponent, use its reciprocal if the final answer requires positive exponents.

Keep coefficients, bases, and exponents in separate mental steps. Then apply any sign from the base, respecting parentheses. If the expression contains sums, simplify or expand them using ordinary algebra before assuming an exponent rule applies. Most errors come from applying a correct rule to the wrong structure.

Short practice

Simplify 4a² × 3a^(−5). The coefficient is 12 and the exponent is 2 + (−5) = −3, so the result is 12/a³ for a ≠ 0. Now simplify (3a²)³: the coefficient is 27 and the exponent is 6, giving 27a⁶. These results use different rules despite containing similar symbols.

Finally compare −2⁴ with (−2)⁴. The first is −16 and the second is 16. Writing the repeated factors, including every parenthesis, is enough to explain the difference. Before submitting an exponent answer, check the base, the operation, and any zero restrictions.