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Exponent — Definition, Rules & Examples

Exponent

x in the expression ax. For example, 3 is the exponent in 23.

 

 

 

See also

Base in an exponential expression, exponent rules

Key Formula

an=a×a×a×⋯×a⏟n factorsa^n = \underbrace{a \times 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

Worked Example

Problem: Evaluate 5^4.
Step 1: Identify the base and exponent. The base is 5 and the exponent is 4.
545^4
Step 2: Write the base as a repeated multiplication, using it as a factor 4 times.
54=5×5×5×55^4 = 5 \times 5 \times 5 \times 5
Step 3: Multiply from left to right: 5 × 5 = 25, then 25 × 5 = 125, then 125 × 5 = 625.
5×5=25,25×5=125,125×5=6255 \times 5 = 25,\quad 25 \times 5 = 125,\quad 125 \times 5 = 625
Answer: 54=6255^4 = 625

Another Example

Problem: Evaluate 3^0 and 4^{-2}.
Step 1: Any nonzero number raised to the exponent 0 equals 1. This is a rule, not a calculation of repeated multiplication.
30=13^0 = 1
Step 2: A negative exponent means you take the reciprocal of the base raised to the positive version of that exponent.
4−2=1424^{-2} = \frac{1}{4^2}
Step 3: Compute 4 squared: 4 × 4 = 16.
4−2=1164^{-2} = \frac{1}{16}
Answer: 30=13^0 = 1 and 4−2=1164^{-2} = \dfrac{1}{16}

Frequently Asked Questions

What does an exponent of 0 mean?
Any nonzero base raised to the power of 0 equals 1. So 70=17^0 = 1, (−3)0=1(-3)^0 = 1, and 1000=1100^0 = 1. The expression 000^0 is a special case that is sometimes left undefined, though in many contexts it is treated as 1.
What does a negative exponent mean?
A negative exponent tells you to take the reciprocal. Specifically, a−n=1ana^{-n} = \frac{1}{a^n} when a≠0a \neq 0. For example, 2−3=123=182^{-3} = \frac{1}{2^3} = \frac{1}{8}. The result is not a negative number — it is a fraction.

Exponent vs. Base

Why It Matters

Exponents let you write very large or very small numbers compactly — scientific notation like 3.0×1083.0 \times 10^8 (the speed of light in m/s) depends on them. They appear throughout algebra, geometry (area and volume formulas), finance (compound interest), and science (population growth, radioactive decay). Understanding exponents is also essential for working with polynomials, logarithms, and higher-level mathematics.

Common Mistakes

Mistake: Multiplying the base by the exponent instead of using repeated multiplication. For example, writing 25=102^5 = 10 instead of 25=322^5 = 32.
Correction: The exponent tells you how many times the base appears as a factor. 252^5 means 2×2×2×2×22 \times 2 \times 2 \times 2 \times 2, not 2×52 \times 5.
Mistake: Thinking a negative exponent makes the result negative. For example, writing 3−2=−93^{-2} = -9.
Correction: A negative exponent produces a reciprocal, not a negative number. 3−2=132=193^{-2} = \frac{1}{3^2} = \frac{1}{9}, which is positive.

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