A 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.
The base is the number that repeats as a factor; the exponent counts the factors. With a negative number, parentheses decide whether the minus sign is part of the base.
Formal definition and notation
In a powerbn, the base b is the factor being multiplied and the exponentn tells how many times it appears: for a positive whole numbern, bn is the product of n factors of b. The base can be any number or expression, such as 10, −2, x or (x + 1). An exponent applies only to the symbol directly before it, unless parentheses group a larger base.
Symbol
Read it as
Meaning
bn
b to the n-th power, or b to the n
the base b used as a factor n times
(−3)2
negative 3, squared
the base is −3, so the value is 9
−32
the opposite of 3 squared
the base is 3, so the value is −9
Key facts
Read the base first: in 103 the base is 10, so 103 = 1000.
A negative base needs parentheses: (−2)4 = 16 but −24 = −16.
In 5x2, only x is squared; the base is x, and 5 is the coefficient. In (5x)2 the base is 5x.
Any nonzero base to the power 0 is 1: 70 = 1. A base to the power 1 is itself: 71 = 7.
In (−5)2 the parentheses make the base −5: (−5)(−5) = 25.
In −52 the exponent touches only the 5, so the base is 5: 52 = 25.
The minus sign is applied last: −52 = −25.
Answer:(−5)2 = 25 and −52 = −25.
Example 3: Same base, add the exponents
Test-style
Write 53 × 52 as a single power of 5, then evaluate it.
Both powers have the same base, 5.
53 × 52 is 3 factors of 5 times 2 more factors of 5: 5 factors in all.
So 53 × 52 = 55.
55 = 5 × 5 × 5 × 5 × 5 = 3125.
Answer:53 × 52 = 55, which equals 3125.
Common mistakes
✗ MistakeIn 34, saying the base is 4.
✓ InsteadIn 34, 3 is the base and 4 is the exponent.
Do not confuse the base with the exponent. The base sits on the line and repeats; the exponent is the small raised number that counts.
✗ MistakeMultiplying the base by the exponent: 34 = 3 × 4 = 12.
✓ Instead34 = 3 × 3 × 3 × 3 = 81.
The exponent tells how many times the base is used as a factor, not what to multiply the base by.
✗ Mistake−32 = 9.
✓ Instead−32 = −9. Only (−3)2 = 9.
Without parentheses the base is 3, and the minus sign is applied after squaring.
✗ MistakeCombining different bases: 23 × 32 = 65.
✓ Instead23 × 32 = 8 × 9 = 72, while 65 = 7776.
You can add exponents only when the bases are the same.
Not to be confused with
Basevs.Base of a shape
In geometry the base is a side or face, as in a triangle‘s area 12bh or a prism’s base. In exponents the base is the factor that repeats.
Basevs.Base ten (number base)
Our place value system is base ten: each place is 10 times the one to its right. That base is the number of digits used (0 to 9), not the base of a power.
The exponent is the raised number that counts the factors. In 62, 6 is the base and 2 is the exponent.
Quick check: 3 questions
Question 1 of 3
What is the base in 73?
Show answer and explanation
Answer: B. The base is the number on the line, 7. Choice A is the exponent, 21 multiplies base by exponent, and 343 is the value of the power, 7 × 7 × 7.
Question 2 of 3
What is the value of (−4)2?
Show answer and explanation
Answer: D. The parentheses make the base −4: (−4)(−4) = 16. Choice A is −42, where the base is only 4, and choices B and C multiply by the exponent.
Question 3 of 3
In 2x3, what is the base of the exponent 3?
Show answer and explanation
Answer: C. An exponent applies only to the symbol right before it, so only x is cubed: 2x3 = 2 × x × x × x. For the base to be 2x you would need (2x)3.
Base: frequently asked questions
What is the difference between a base and an exponent?
The base is the number being multiplied; the exponent says how many times it is used as a factor. In 53, the base 5 is written normal size and the exponent 3 is written small and raised. So 53 = 5 × 5 × 5 = 125. Swapping them changes the value: 35 = 243.
Can the base be negative or a fraction?
Yes. Write a negative or fraction base in parentheses: (−2)3 = −8 and (12)3 = 18. A negative base gives a positive result with an even exponent and a negative result with an odd exponent. A fraction base between 0 and 1 gets smaller as the exponent grows.
What does base mean in other parts of math?
The word has three common meanings. In exponents it is the repeated factor, as on this page. In geometry it is a side or face that a height is measured from, as in the base of a triangle or prism. In place value it is the size of the number system: base ten uses digits 0 to 9, and base two uses 0 and 1.
PowerA 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 7ExponentAn 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 8Whole NumberA whole number is 0 or a positive counting number: 0, 1, 2, 3, and so on. Whole numbers do not include fractions, decimals, or negative numbers, so 12 is a whole number but 8.5, 34 and −3 are not. Every whole number is also an integer.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 2CoefficientA coefficient is the number multiplied by a variable in a term. In 4a the coefficient is 4, in −3x it is −3, and in x it is 1, because x = 1x. The sign in front belongs to the coefficient. A number standing alone, with no variable, is a constant instead.Algebra · Grade 6, Grade 7, Grade 8Scientific 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 1EvaluateEvaluate means to find the single number value of an expression. Replace each variable with the number you are given, then calculate using the order of operations. For example, 2x + 7 evaluated at x = 3 is 2(3) + 7 = 13. The result is one number, not a new expression.Algebra · Grade 6, Grade 7, Algebra 1SideA 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 2TriangleA triangle is a polygon with three sides and three angles. Its three corners are called vertices, and the angle measures in any triangle add to 180 degrees. Triangles are classified by their sides (equilateral, isosceles, scalene) and by their angles (acute, right, obtuse).Geometry · Grade 3, Grade 4, Grade 5Place ValuePlace value is the value of a digit based on its position in a number. In 3,507 the 5 is worth 500, because it sits in the hundreds place. Each place is worth 10 times the place to its right: ones, tens, hundreds, thousands, and after the decimal point, tenths and hundredths.Number System · Grade 3, Grade 4, Grade 5HeightHeight is how tall something is: the distance from its bottom to its top, measured straight up and down. In geometry, the height of a shape is the perpendicular distance from a base to the opposite vertex or side, so it always meets the base at a right angle. Area and volume formulas use it.Measurement · Grade 2, Grade 5, Grade 6MultiplicationMultiplication 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 4PolynomialA polynomial is an expression built from variables and coefficients using only addition, subtraction, and multiplication, with whole-number exponents. Examples are 3x2 + 5x − 7 and x + 9. Its degree is the highest exponent it contains. A variable in a denominator or under a square root usually means the expression is not a polynomial.Algebra · Algebra 1, Algebra 2Laws of ExponentsThe 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.Algebra · Grade 8, Algebra 1, Algebra 2Exponential 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, PrecalculusSquare RootA square root of a number is a value that, multiplied by itself, gives that number. The square root of 36 is 6 because 6 × 6 = 36. Every positive number has two square roots, such as 6 and −6, but the radical sign √ means the principal square root, which is the non-negative one.Number System · Grade 8, Algebra 1