Laws of Exponents
Apply integer-exponent rules for nonzero bases: add exponents when multiplying matching-base powers, subtract when dividing, multiply when raising a power to a power, and interpret zero and negative exponents (3² × 3⁻⁵ = 3⁻³ = 1/27).
What a learner can do afterwards
- Simplify products and quotients of powers with the same nonzero base, and powers raised to powers (e.g., 5³ × 5⁴ = 5⁷, 2⁸ ÷ 2³ = 2⁵, (3²)⁴ = 3⁸)
- Evaluate expressions with zero and negative exponents (e.g., 7⁰ = 1, 2⁻³ = 1/8)
- Spot and correct a common exponent error, such as claiming 3² × 3³ = 9⁵
1 · Read
You have used powers to count repeated factors. In 3² = 3 × 3, the base is 3 and the exponent 2 counts factors. With matching bases, keep the base and add exponents: 3² × 3⁴ = 3⁶. Do not multiply the matching bases. A letter such as n, x, y or p can be a base too. We write the product rule as aᵐ × aⁿ = aᵐ⁺ⁿ, where a is the base and m and n are exponents.
For division with matching nonzero bases, keep the base and subtract exponents: aᵐ ÷ aⁿ = aᵐ⁻ⁿ. For example, 8⁵ ÷ 8² = 8³ because 5 − 2 = 3. Cancelling two factors of 8 from both numerator and denominator leaves three factors of 8. Check the bases and operation first. The product and quotient rules do not apply to unlike bases or to addition.
A power raised to a power repeats the whole inner power. For example, (2²)⁴ has four factors of 2². Each contains two factors of 2, so there are 2 × 4 = 8 factors of 2 altogether: (2²)⁴ = 2⁸. For (aᵐ)ⁿ, multiply m and n to get the new exponent. Keep the base a.
Any nonzero base raised to zero equals 1: a⁰ = 1 for a ≠ 0. The sign ≠ means 'is not equal to'. For example, 9³ ÷ 9³ = 1, and subtracting exponents gives 9⁰. The rule does not apply to 0⁰. Check the whole base inside any brackets. In 5 × 2⁰, only 2 is raised to zero, so the result is 5 × 1 = 5.
For a nonzero base a and a positive integer n, a⁻ⁿ = 1/aⁿ. A negative exponent means the reciprocal of the positive power. For example, 4⁻² = 1/4² = 1/16. It does not put a minus sign in front of the answer. To compare negative powers of positive bases, evaluate their fractions. Among fractions 1/d with positive denominators, the larger denominator gives the smaller fraction.
For a nonzero base a, combine integer exponents from left to right: add for × and subtract for ÷. Adding −2 means subtracting 2. Subtracting −2 means adding 2: 6² ÷ 6⁻² = 6⁴. For a negative result −k, use a⁻ᵏ = 1/aᵏ, where k is a positive integer. Multiply k factors of a to find the denominator. For example, 6³ × 6⁻² ÷ 6² gives 3 + (−2) − 2 = −1, so the result is 6⁻¹ = 1/6.
For a nonzero base, add exponents to multiply, subtract to divide, and multiply for a power of a power. A zero exponent gives 1; a negative exponent gives a reciprocal.
Original lesson, practice questions and flashcards written for this dataset.
Original readings, practice questions and flashcards written for this dataset. Linked videos are credited to Khan Academy in the source records.
More to read
Count factors in two different expressions
Compare 6² × 6⁴ with (6²)⁴. In the product, one power contributes two factors of 6 and the other contributes four. There are six factors altogether, giving 6⁶.
The brackets change the task. In (6²)⁴, the outer exponent asks for four copies of 6². Each copy has two factors of 6, giving eight factors altogether: 6⁸.
Look at the operation before choosing a rule. A product of matching-base powers adds exponents. A power raised to a power multiplies them. Both calculations keep the base 6.
What cancellation leaves
Write 11⁷ ÷ 11⁴ as a fraction. The numerator has seven factors of 11, and the denominator has four. Cancel four matching factors from each. Three factors remain in the numerator, so the result is 11³. The exponent calculation is 7 − 4 = 3.
If both powers are 11⁴, all their factors cancel and the quotient is 1. Subtracting exponents also gives 11⁰, so 11⁰ = 1.
These rules require a nonzero base. Check brackets before applying the zero-exponent rule. In 2 × 11⁰, only 11 has exponent zero, so the value is 2 × 1 = 2.
Continue below exponent zero
Start with 5² = 25. Lowering the exponent by one divides the value by 5: 5¹ = 5, then 5⁰ = 1. Continue dividing and you get 5⁻¹ = 1/5 and 5⁻² = 1/25. The negative exponent produces a reciprocal, not a negative value.
One fifth is larger than one twenty-fifth. For fractions with numerator 1 and positive denominators, a larger denominator means a smaller fraction.
You can also combine powers before evaluating. For 8² × 8⁻⁴, add 2 and −4 to get −2. Then 8⁻² = 1/8² = 1/64. Negative powers require a nonzero base.
Explore more
2 · Watch
به انگلیسیTake it off screen
Where it sits
Learn first
Optional connections
This opens up
Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
14 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.