The laws of exponents are the rules for simplifying powers that share a base or an exponent. Multiply powers of the same base by adding exponents, divide by subtracting them, and raise a power to a power by multiplying them: x2 · x5 = x7, x6x2 = x4, (x3)2 = x6.
In Spanish
leyes de los exponentes
Also called
exponent rules, properties of exponents, exponent laws, rules of exponents, index laws
Green cards combine powers of the same base; blue cards spread one exponent over a product, a quotient or another power; yellow cards extend exponents to 0, negatives and fractions.
Formal definition and notation
For real numbersa, b ≠ 0 and integers m, n: aman = am + n, aman = am − n, (am)n = amn, (ab)n = anbn, ( ab )n = anbn, a0 = 1 and a−n = 1an. The same laws extend to rational exponents when a, b > 0. They follow from the meaning of a positive integerexponent as repeated multiplication.
the common base (nonzero when it is divided or has a zero or negative exponent)
m, n
the exponents
(ab)n = anbn
an exponent outside parentheses goes to every factor inside
The product and quotient laws need the same base. 23 · 53 can use the power-of-a-product law in reverse, (2 · 5)3 = 103, but 23 · 52 has no shortcut.
Try it: count the factors
Interactive
Set the base to 3 and compare 32, 33 and 35: the tiles show that 32 · 33 has 2 + 3 = 5 factors, which is the product law. Step the exponent below 0 to see the negative exponent law.
This interactive needs JavaScript. The worked examples below show the same ideas step by step.
Negative exponent law: b−3 = 1b3, so the result is a4b3.
Answer: The expression simplifies to a^4/b^3, that is a4b3.
Example 3: Solve for an exponent
Test-style
If 4x · 26 = 220, what is x?
Write 4 as a power of 2: 4x = (22)x = 22x.
Product law: 22x · 26 = 22x + 6.
Same base on both sides, so the exponents match: 2x + 6 = 20.
Solve: 2x = 14, so x = 7.
Answer: x = 7
Common mistakes
✗ Mistakex3 · x4 = x12.
✓ Insteadx3 · x4 = x7.
Multiplying powers adds exponents: 3 factors of x times 4 more factors is 7 factors. Multiplying exponents is the rule for a power of a power, (x3)4 = x12.
✗ Mistake32 · 34 = 96.
✓ Instead32 · 34 = 36 = 729.
The base stays the same; only the exponents combine. 96 is 531,441, far too big.
✗ Mistake(x + 2)2 = x2 + 4.
✓ Instead(x + 2)2 = x2 + 4x + 4.
The power-of-a-product law works for products, not sums. Expand (x + 2)(x + 2) instead.
✗ Mistake(3y)2 = 3y2.
✓ Instead(3y)2 = 9y2.
An exponent outside parentheses applies to every factor inside, so the 3 is squared too.
Order of operations tells you when to evaluate a power; exponent laws let you rewrite powers without evaluating them. 23 · 22 = 25 skips computing 8 and 4.
Quick check: 3 questions
Question 1 of 3
Simplify y9y3.
Show answer and explanation
Answer: C. Quotient law: subtract the exponents, 9 − 3 = 6. y3 divides the exponents, y12 adds them, and y27 multiplies them.
Question 2 of 3
Which is equal to (2x5)3?
Show answer and explanation
Answer: A. Cube each factor: 23 = 8 and (x5)3 = x15. 6x15 multiplies 2 by 3, 2x15 forgets to cube the 2, and 8x8 adds the exponents.
Question 3 of 3
What is 53 · 5−5?
Show answer and explanation
Answer: D. Add exponents: 3 + (−5) = −2, and 5−2 = 125. −25 treats the negative exponent as a negative value, and 5−15 multiplies the exponents.
Laws of Exponents: frequently asked questions
What are the 7 laws of exponents?
The usual list is: product law aman = am + n, quotient law aman = am − n, power of a power (am)n = amn, power of a product (ab)n = anbn, power of a quotient, zero exponent a0 = 1 and negative exponent a−n = 1an. Many books add an eighth for rational exponents.
When do you add exponents and when do you multiply them?
Add exponents when you multiply two powers with the same base: x2 · x3 = x5. Multiply exponents when one power is raised to another power: (x2)3 = x6. A quick test: if there are parentheses with an exponent outside, multiply; if two powers sit side by side, add.
Can you use exponent laws with different bases?
Only when the exponents match, using the power-of-a-product law backward: 24 · 54 = 104. Otherwise, try rewriting one base as a power of the other, such as 8 = 23 or 9 = 32. If neither works, as in 23 · 32, just evaluate: 8 · 9 = 72.
Updated November 2026·Reviewed by the Effortless Math team·Standards: CCSS 8.EE.A.1, CCSS HSN-RN.A.2
Cite this entry
Effortless Math Team. “Laws of Exponents.” Effortless Math Dictionary. Updated November 2026. https://www.effortlessmath.com/math-dictionary/laws-of-exponents/
Flashcard
TermLaws of Exponentsadd, subtract or multiply the exponents?
DefinitionSame base: multiply powers by adding exponents, divide by subtracting. Power of a power: multiply exponents. Example: x2 · x3 = x5, (x2)3 = x6.
Real NumberA real number is any number that can be placed on the number line. The real numbers are all the rational numbers (fractions, integers, terminating and repeating decimals) together with all the irrational numbers (such as √2 and π). Square roots of negative numbers, like √−4, are not real.Number System · Grade 8, Algebra 1Rational ExponentA rational exponent is an exponent written as a fraction, and it stands for a root. The denominator is the root and the numerator is the power: a1n = n√a and amn = ( n√a )m. For example, 912 = 3 and 2723 = 32 = 9.Algebra · Algebra 1, Algebra 2Positive IntegerA positive integer is a whole number greater than zero: 1, 2, 3, 4 and so on. Positive integers have no fraction or decimal part and are never 0 or negative. They are also called counting numbers, and the set is written ℤ+. When a test says n is a positive integer, n cannot be 0, −3 or 2.5.Problem Solving · Grade 6, Grade 7, Algebra 1ExponentAn exponent tells how many times to use a base as a factor. It is the small raised number to the upper right of the base: in 53, the base is 5 and the exponent is 3, so 53 = 5 × 5 × 5 = 125. Any nonzero number with an exponent of 0 equals 1.Algebra · Grade 6, Grade 7, Grade 8MultiplicationMultiplication is the operation that combines equal groups to find a total. The fact 4 × 6 = 24 means 4 groups of 6, or 6 + 6 + 6 + 6. The numbers you multiply are called factors, and the result is the product. Order does not matter: 6 × 4 is also 24.Operations · Grade 2, Grade 3, Grade 4PowerA power is a number written as a base with an exponent, such as 53, or the value that results, 53 = 5 × 5 × 5 = 125. We say 125 is the third power of 5, or “5 to the third power”. The second power is called the square and the third power the cube.Number System · Grade 5, Grade 6, Grade 7QuotientA quotient is the answer you get when you divide one number by another. In 20 ÷ 4 = 5, the quotient is 5. The number being divided is the dividend and the number you divide by is the divisor. When the division is not exact, the quotient comes with a remainder, or as a decimal or fraction.Operations · Grade 3, Grade 4, Grade 5Zero ExponentA zero exponent is an exponent of 0, and any nonzero number raised to the power 0 equals 1. So 70 = 1, (−3)0 = 1 and x0 = 1 for x ≠ 0. The rule keeps the quotient rule working: 5353 = 50, and a number divided by itself is 1.Algebra · Grade 8, Algebra 1Negative ExponentA negative exponent tells you to take the reciprocal of the base raised to the matching positive exponent: a−n = 1an for a ≠ 0. So 2−3 = 18 and 10−2 = 0.01. A negative exponent makes a number smaller in size, but it never makes the value negative.Algebra · Grade 8, Algebra 1Scientific NotationScientific notation writes a number as a value from 1 to less than 10 times a power of 10, such as 5.4 × 103 for 5,400. The exponent records how far the decimal point moved: positive for large numbers and negative for numbers between 0 and 1, as in 0.0062 = 6.2 × 10−3.Number System · Grade 8, Algebra 1SimplifySimplify means to rewrite a number or expression in a shorter, easier form without changing its value. To simplify a fraction, divide the numerator and denominator by a common factor until none is left: 1824 = 34. To simplify an expression, combine like terms and do the operations you can: 3x + 2x = 5x.Number System · Grade 4, Grade 5, Grade 6NumeratorThe numerator is the top number in a fraction. It tells how many parts are being counted, while the denominator below the fraction bar tells how many equal parts make one whole. In 35, the numerator is 3: three of five equal parts are taken.Number System · Grade 3, Grade 4, Grade 5NumberA number is an amount or value used to count, measure, order and compare. We write numbers with digits (7), words (seven) or other symbols, but the number itself is the amount: 7 dots, the word seven and the numeral 7 all show the same number. Numbers include whole numbers, fractions, decimals and negatives.Number System · Kindergarten, Grade 1, Grade 2SideA side is one of the straight line segments that form the outline of a polygon. Two sides meet at each corner, or vertex. A triangle has 3 sides and a square has 4, and adding the lengths of all the sides gives the perimeter.Geometry · Kindergarten, Grade 1, Grade 2Properties of LogarithmsThe properties of logarithms are rules for rewriting logs: the log of a product is a sum, logb(MN) = logbM + logbN; the log of a quotient is a difference; and the log of a power is a multiple, logb(Mp) = plogbM. They come from the rules of exponents.Functions · Algebra 2, PrecalculusLogarithmA logarithm is the exponent you need to raise a base to in order to get a given number. logbx = y means by = x, so log2 32 = 5 because 25 = 32. A log with no base written uses base 10: log 1000 = 3. Logs undo exponents.Functions · Algebra 2, PrecalculusOrder of OperationsThe order of operations is the agreed sequence for evaluating an expression: parentheses first, then exponents, then multiplication and division from left to right, then addition and subtraction from left to right. Without it the same expression would have several answers. Many students remember it as PEMDAS.Algebra · Grade 5, Grade 6, Grade 7BaseA base is the number or expression that is repeatedly multiplied when using an exponent. In 45 the base is 4 and the exponent is 5, so 45 = 4 × 4 × 4 × 4 × 4 = 1024. The base is the big number written on the line; the exponent is the small raised number.Algebra · Grade 6, Grade 7, Grade 8MonomialA monomial is a single term made of a number, a variable, or a product of numbers and variables with whole-number exponents, such as 8, x, or −7x2y. It has no addition or subtraction, and no variable in a denominator. Its degree is the sum of the exponents on its variables.Algebra · Grade 8, Algebra 1Exponential FunctionAn exponential function is a function in which the input is the exponent: f(x) = a · bx, with a ≠ 0, base b > 0 and b ≠ 1. Each time x goes up by 1, the output is multiplied by the same factor b. If b > 1 the function grows; if 0 < b < 1 it decays.Functions · Algebra 1, Algebra 2, Precalculus