Apply the properties of integer exponents, including zero and negative exponents, to generate equivalent expressions

Learning goal: Understand, explain and apply the properties of integer exponents, including zero and negative exponents, to generate equivalent numerical expressions and to justify each step.

A power such as 3⁴ is shorthand for repeated multiplication: 3⁴ = 3 × 3 × 3 × 3 = 81. The 3 is the base and the 4 is the exponent, and for a positive integer exponent n, bⁿ means n factors of b multiplied together. Every property in this lesson comes from counting factors, and every extension to zero and negative exponents is chosen so that the same counting rules keep working. When a rule seems mysterious, expand a small case into its factors and count.

Product of powers. 3² × 3⁵ = (3 × 3) × (3 × 3 × 3 × 3 × 3) = 3⁷. Two factors joined with five factors make seven factors, so bᵐ × bⁿ = bᵐ⁺ⁿ (for any base when both exponents are positive, and for a nonzero base once an exponent is zero or negative, because those exponents involve dividing by the base). The bases must be the same; 3² × 5³ cannot be combined into a single power because the factors are different numbers. The exponents are added, not multiplied: 3² × 3⁵ is 3⁷ = 2187, not 3¹⁰.

Quotient of powers. 3⁵ ÷ 3² = (3 × 3 × 3 × 3 × 3)/(3 × 3). Two of the factors cancel, leaving 3³ = 27, so bᵐ ÷ bⁿ = bᵐ⁻ⁿ for a nonzero base b. The base must be nonzero because division by zero is undefined.

Power of a power. (3²)⁴ = 3² × 3² × 3² × 3², which is four groups of two factors, or eight factors, so (bᵐ)ⁿ = bᵐⁿ (again with a nonzero base whenever an exponent is zero or negative). Here the exponents multiply. Students often confuse this with the product rule; the check is to ask whether the expression is a product of powers (add) or a power raised to a power (multiply).

Power of a product and of a quotient. (2 × 5)³ = (2 × 5)(2 × 5)(2 × 5) = 2³ × 5³ = 8 × 125 = 1000, and directly (10)³ = 1000. In general (ab)ⁿ = aⁿbⁿ, with a and b both nonzero when n is zero or negative, and (a/b)ⁿ = aⁿ/bⁿ with b nonzero for every n and a nonzero as well when n is zero or negative, because a zero or negative exponent involves dividing by the base. This rule distributes an exponent over multiplication or division only; it does not distribute over addition. (2 + 5)² = 49, while 2² + 5² = 29.

The zero exponent. What should 3⁰ mean? The quotient rule gives 3³ ÷ 3³ = 3³⁻³ = 3⁰, and 3³ ÷ 3³ is 27/27 = 1. So if the quotient rule is to keep working, 3⁰ must equal 1. The same argument works for any nonzero base, so b⁰ = 1 whenever b ≠ 0. The expression 0⁰ is not given a value in this course, because the argument that produced b⁰ = 1 divided by b, and dividing by zero is not allowed. The table in this lesson shows the same idea as a pattern: going down from 3³ = 27 to 3² = 9 to 3¹ = 3 divides by 3 each time, so the next entry, 3⁰, is 3 ÷ 3 = 1.

Negative exponents. Continue the pattern one more step: 3⁰ = 1, so 3⁻¹ should be 1 ÷ 3 = 1/3, then 3⁻² = 1/9 and 3⁻³ = 1/27. The quotient rule says the same thing: 3² ÷ 3⁵ = 3²⁻⁵ = 3⁻³, and by canceling factors directly, (3 × 3)/(3 × 3 × 3 × 3 × 3) = 1/(3 × 3 × 3) = 1/27. So the definition is b⁻ⁿ = 1/bⁿ for nonzero b, and equivalently 1/b⁻ⁿ = bⁿ. A negative exponent means a reciprocal; it does not by itself make the value negative: 3⁻² = 1/9, which is positive. A negative exponent only takes a reciprocal, so the sign of b⁻ⁿ is the sign of bⁿ. For a positive base the value is always positive. For a negative base the parity of the exponent decides: (−2)⁻² = 1/(−2)² = 1/4 is positive because the exponent is even, and (−2)⁻³ = 1/(−2)³ = 1/(−8) = −1/8 is negative because the base is negative and the exponent is odd. And 0⁻ⁿ is undefined, because it would mean 1/0ⁿ = 1/0.

The example that ties the rules together. Simplify 3² × 3⁻⁵. Product rule: the exponents add, giving 3²⁺⁽⁻⁵⁾ = 3⁻³. Negative exponent rule: 3⁻³ = 1/3³ = 1/27. Each step is an equivalent numerical expression, so 3² × 3⁻⁵ = 3⁻³ = 1/3³ = 1/27, and a direct check confirms it: 3² = 9, 3⁻⁵ = 1/243, and 9/243 = 1/27.

A fully worked model with several rules. Write (2³ × 2⁻⁵)² ÷ 2⁻⁶ as a single power and evaluate. Inside the parentheses, 2³ × 2⁻⁵ = 2⁻². Raising to the second power, (2⁻²)² = 2⁻⁴. Dividing by 2⁻⁶ subtracts the exponent: 2⁻⁴ ÷ 2⁻⁶ = 2⁻⁴⁻⁽⁻⁶⁾ = 2². The value is 4. Verify numerically: 2³ × 2⁻⁵ = 8 × 1/32 = 1/4; squared is 1/16; dividing by 2⁻⁶ = 1/64 means multiplying by 64, and 64/16 = 4. The two routes agree, which is what "equivalent expressions" means.

Negative bases and where the sign lives. (−2)⁴ = (−2)(−2)(−2)(−2) = 16, because the parentheses make −2 the base. But −2⁴ means −(2⁴) = −16, because the exponent applies only to 2 and the negative sign is applied afterward. Similarly (−2)³ = −8: an odd number of negative factors gives a negative product. Read the parentheses before applying any rule.

Fractions as bases. (2/3)⁻² = 1/(2/3)² = 1/(4/9) = 9/4. A quicker route, for a fraction whose numerator and denominator are both nonzero, is that a negative exponent flips the fraction: (2/3)⁻² = (3/2)² = 9/4. Both routes are the quotient-power rule combined with the negative-exponent rule.

Common errors and why they are wrong. Writing 3⁻² = −9 treats the exponent as a sign; the exponent −2 means "one over 3²", so the value is 1/9. Writing 2³ × 2⁴ = 4⁷ multiplies the bases; only the exponents combine, and the base stays 2, so the answer is 2⁷ = 128, whereas 4⁷ = 16384. Writing (3²)³ = 3⁵ adds where it should multiply; the correct result is 3⁶ = 729, and expanding to three groups of two factors shows why. Writing 5⁰ = 0 ignores the quotient argument; 5⁰ = 1. Each of these errors is exposed by expanding a small case into factors and counting.

Worked quotient with a negative exponent: Simplify 2⁵ ÷ 2⁸. Subtract exponents: 2⁵⁻⁸ = 2⁻³ = 1/2³ = 1/8. Check by canceling: five factors of 2 over eight factors of 2 leaves three factors of 2 in the denominator, so the result is 1/8. The negative exponent signals that more factors remained below the fraction bar than above it.

Worked power of a power with a negative exponent: Simplify (5⁻¹)² and (5²)⁻¹. Both multiply the exponents: (5⁻¹)² = 5⁻² = 1/25, and (5²)⁻¹ = 5⁻² = 1/25. Numerically, (1/5)² = 1/25 and 1/25 again. The order of the two operations does not matter because the exponents multiply either way.

Worked mixed bases: Simplify (2 × 3)⁴ ÷ (2² × 3⁵). Distribute the exponent: 2⁴ × 3⁴. Then divide power by power with matching bases: 2⁴⁻² × 3⁴⁻⁵ = 2² × 3⁻¹ = 4 × 1/3 = 4/3. Check: 6⁴ = 1296 and 2² × 3⁵ = 4 × 243 = 972, and 1296/972 = 4/3. The bases 2 and 3 are handled separately because the product and quotient rules require a common base.

Worked zero exponent inside an expression: Evaluate 7⁰ + 4 × 2⁻³ − (−1)⁴. 7⁰ = 1 since 7 ≠ 0; 2⁻³ = 1/8, so 4 × 1/8 = 1/2; (−1)⁴ = 1 because four negative factors give a positive product. The total is 1 + 1/2 − 1 = 1/2.

Property record: Make a table with one row per property: product, quotient, power of a power, power of a product, zero exponent, negative exponent. In each row write the rule with letters, a numerical example with base 3, the same example expanded into factors, and the condition under which the rule holds (for example, base nonzero). Then extend the powers-of-3 table from 3⁴ down to 3⁻⁴, writing beside each step "divide by 3".

Independent performance: Create six numerical expressions of your own that each require at least two properties, including one with a negative base in parentheses, one with a fraction base and a negative exponent, and one that simplifies to a zero exponent. Simplify each to a single power and then to a number, naming the property used at every step. Verify two of them by computing the powers directly and comparing. Success criteria: every step names its property, the two verified expressions agree by both routes, and no negative exponent is left in a final answer.