An 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.
The exponent counts how many times the base appears as a factor. Lowering the exponent by 1 divides by the base, which is why 20 is 1 and 2−1 is one half.
Formal definition and notation
For a numbera and a whole numbern ≥ 1, an is the product of n factors, each equal toa. The number a is the base and n is the exponent; the whole expression an is a power of a. For a ≠ 0, the definition extends to zero and negative integers by a0 = 1 and a−n = 1an, which keeps the rule am × an = am + n true for all integers.
Powers of 10 shift the decimal point and power scientific notation: 103 = 1,000 and 10−2 = 0.01.
Exponent definition and rules
Exponent definition and rules
an = a × a × ⋯ × a (n factors)
a
the base: the number being multiplied
n
the exponent: how many factors of the base
am × an = am + n
same base, multiply: add the exponents
(am)n = amn
power of a power: multiply the exponents
a0 = 1, a−n = 1an
zero and negative exponents, for a ≠ 0
The rules work only when the bases match. 23 × 24 = 27, but 23 × 54 cannot be combined into one power.
Try it: build a power
Interactive
Change the base and the exponent. The tiles show each factor of the base, and the readout gives the product. Try an exponent of 0, a negative exponent, and a negative base with an even and then an odd exponent.
This interactive needs JavaScript. The worked examples below show the same ideas step by step.
Exponent examples, step by step
Example 1: Evaluate a power
Warm-up
Evaluate 63.
The base is 6 and the exponent is 3, so write three factors of 6: 6 × 6 × 6.
Multiply from left to right: 6 × 6 = 36.
Then 36 × 6 = 216.
Answer:63 = 216
Example 2: Where the negative sign sits
Core
Evaluate (−2)4 and −24.
In (−2)4 the parentheses make the base −2: (−2)(−2)(−2)(−2).
Four negative factors give a positive product: 4 × 4 = 16.
In −24 the base is only 2. The exponent comes first: 24 = 16.
Then apply the negative sign: −16.
Answer:(−2)4 = 16 but −24 = −16.
Example 3: Repeated doubling
Test-style
A lab culture starts with 50 cells, and the number of cells doubles every hour. How many cells are there after 6 hours?
Each hour multiplies the count by 2, so 6 hours multiply it by 2 six times: 26.
26 = 2 × 2 × 2 × 2 × 2 × 2 = 64.
Multiply by the starting count: 50 × 26 = 50 × 64 = 3,200.
Trap: 50 × 2 × 6 = 600 treats the exponent as an ordinary factor.
Answer: There are 3,200 cells after 6 hours.
Common mistakes
✗ Mistake34 = 3 × 4 = 12
✓ Instead34 = 3 × 3 × 3 × 3 = 81
Do not multiply the base by the exponent. The exponent counts how many 3s you multiply, so 3 is used as a factor four times.
✗ Mistake−32 = 9
✓ Instead−32 = −(3 × 3) = −9, while (−3)2 = 9.
An exponent applies only to what sits directly to its left. Without parentheses, the base is 3 and the minus sign is applied afterward.
✗ Mistake70 = 0
✓ Instead70 = 1
Each step down in the exponent divides by 7: 71 = 7, so 70 = 7 ÷ 7 = 1. A zero exponent never makes the answer 0 for a nonzero base.
✗ Mistake23 × 24 = 212
✓ Instead23 × 24 = 23 + 4 = 27 = 128
Three 2s times four 2s is seven 2s in all. Add the exponents when you multiply powers of the same base; multiply them only for a power of a power.
A square root undoes the exponent 2. 62 = 36, and √36 = 6.
Quick check: 3 questions
Question 1 of 3
What is the value of 25?
Show answer and explanation
Answer: B.25 = 2 × 2 × 2 × 2 × 2 = 32. 10 multiplies the base by the exponent, 25 is 52 (base and exponent swapped), and 7 adds them.
Question 2 of 3
Which expression equals 1?
Show answer and explanation
Answer: C. Any nonzero number to the power 0 is 1, so 70 = 1. 71 is 7, 07 is 0 (seven factors of 0), and 7 × 0 is 0.
Question 3 of 3
Simplify x3 × x4.
Show answer and explanation
Answer: D. Three factors of x times four factors of x is seven factors, so add the exponents: x7. x12 multiplies the exponents, and the choices with a 2 in front add the bases as well.
Exponent: frequently asked questions
Why does any number to the power of 0 equal 1?
Look at the pattern 33 = 27, 32 = 9, 31 = 3. Each time the exponent drops by 1, the value is divided by 3. One more step gives 30 = 3 ÷ 3 = 1. The rule a0 = 1 holds for every nonzero base. The expression 00 is left undefined in most school courses.
What does a negative exponent mean?
A negative exponent means take the reciprocal of the positive power. 5−2 = 152 = 125. It does not make the answer negative: 5−2 is a small positive number. Negative exponents appear in scientific notation for tiny quantities, such as 3 × 10−4 = 0.0003.
Is an exponent the same as a power?
People use the words loosely. Many teachers say an exponent tells you how many times to multiply a number by itself, and call 3 in 23 the power. Strictly, the exponent is the small number 3, and the power is the whole expression 23 or its value, 8. Either way, 23 is read two to the third power.
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