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Power

Power

The result of raising a base to an exponent. For example, 8 is a power of 2 since 8 is 23.

 

 

 

See also

Power rule, power series, binomial theorem

Key Formula

an=a×a×⋯×a⏟n factorsa^n = \underbrace{a \times a \times \cdots \times a}_{n \text{ factors}}
Where:
  • aa = The base — the number being multiplied repeatedly
  • nn = The exponent — how many times the base appears as a factor
  • ana^n = The power — the resulting value

Worked Example

Problem: Find the value of 3 raised to the 4th power.
Step 1: Write the expression using exponential notation.
343^4
Step 2: Expand the exponent as repeated multiplication. The base 3 appears 4 times.
34=3×3×3×33^4 = 3 \times 3 \times 3 \times 3
Step 3: Multiply from left to right.
3×3=9,9×3=27,27×3=813 \times 3 = 9, \quad 9 \times 3 = 27, \quad 27 \times 3 = 81
Answer: 34=813^4 = 81, so 81 is a power of 3.

Another Example

Problem: List the first five powers of 2.
Step 1: Compute 212^1 through 252^5 by repeated multiplication.
21=2,22=4,23=8,24=16,25=322^1 = 2, \quad 2^2 = 4, \quad 2^3 = 8, \quad 2^4 = 16, \quad 2^5 = 32
Step 2: Each result is obtained by multiplying the previous one by 2. Notice the pattern: every power of 2 is double the one before it.
Answer: The first five powers of 2 are 2, 4, 8, 16, and 32.

Frequently Asked Questions

Is the 'power' the same thing as the exponent?
Not exactly. The exponent is the small raised number that tells you how many times to multiply the base. The power is the result of that operation. However, in casual usage people often say 'raise to a power' when they mean 'raise to an exponent,' so the two words are sometimes used interchangeably. Strictly, in 23=82^3 = 8, the exponent is 3 and the power is 8.
What is any number raised to the power of 0?
Any nonzero number raised to the 0th exponent equals 1. For example, 50=15^0 = 1 and (−3)0=1(-3)^0 = 1. The expression 000^0 is typically left undefined or defined as 1 depending on context.

Power vs. Exponent

The exponent is the small number written above and to the right of the base (e.g., the 3 in 232^3). The power is the value that results from the calculation (e.g., 8). People often use 'power' loosely to mean 'exponent,' as in 'raise 2 to the 3rd power,' but technically the power refers to the outcome, not the indicator.

Why It Matters

Powers appear throughout mathematics and science. Area uses the second power (s2s^2), volume uses the third power (s3s^3), and compound interest relies on raising a growth factor to the nnth power. Understanding powers is also essential for working with scientific notation, where very large or very small numbers are expressed as a coefficient times a power of 10.

Common Mistakes

Mistake: Confusing ana^n with a×na \times n — for example, writing 23=62^3 = 6 instead of 23=82^3 = 8.
Correction: Exponentiation means repeated multiplication, not multiplication by the exponent. 23=2×2×2=82^3 = 2 \times 2 \times 2 = 8, which is different from 2×3=62 \times 3 = 6.
Mistake: Ignoring parentheses with negative bases, such as writing −24=16-2^4 = 16.
Correction: Without parentheses, the exponent applies only to 2, so −24=−(24)=−16-2^4 = -(2^4) = -16. If you want the base to be negative, write (−2)4=16(-2)^4 = 16.

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