Exponents — Definition, Laws, and Examples
Exponents — Definition, Laws, and Examples
What Are Exponents?
An exponent is a small raised number that tells you how many times to use the base in a repeated multiplication. In bnb^n, the base b is the number being multiplied and the exponent n (also called the power or index) is the count of factors.
bn=b×b×⋯×b⏟n times
So 34=3×3×3×3=81. The base is 3, the exponent is 4, and we say "3 to the fourth power" or "3 raised to the power 4." Two powers have their own spoken names: b2 is "b squared" and b3 is "b cubed." Those names come from area and volume — a square of side b has area b2, a cube of side b has volume b3.
The Parts of a Power
| Term | What it is | In 535^3 |
|---|---|---|
| Base | The number being multiplied | 5 |
| Exponent | How many times the base is used | 3 |
| Value | The result of the multiplication | 125 |
A small distinction worth getting right early: −2^4 and (−2)^4 are not the same. Without brackets, the exponent attaches only to the 2, so −2^4 = −(2^4) = −16. With brackets, the whole −2 is the base, so (−2)^4 = 16. That bracket is responsible for more lost marks than almost any other exponent slip.
What Are the Seven Laws of Exponents?
The laws of exponents are shortcuts that let you combine and simplify powers without expanding them. Here they are at a glance — each with a one-line reason and a link to the article that works it in full depth.
| Law | Rule | Quick example |
|---|---|---|
| Product of powers | a^m⋅a^n = a^{m+n} | 2^3⋅2^4 = 2^7 |
| Quotient of powers | a^m/a^n = a^{m-n} | 5^6/5^2 = 5^4 |
| Power of a power | (a^m)^n = a^{mn} | (3^2)^4 = 3^8 |
| Power of a product | (ab)^m = a^m b^m | (2x)^3 = 8x^3 |
| Power of a quotient | (a/b)^m = a^m/b^m | (2/3)^2 = 4/9 |
| Zero exponent | a^0 = 1; (a ≠ 0) | 7^0 = 1 |
| Negative exponent | a^(-n) = 1/a^n | 4^(-2) = 1/16 |
Each law follows from the definition, not from memorisation.
Why Is the Zero Exponent Equal to One?
Look at the quotient rule with equal exponents:
a^3/a^3 = a^{3-3} = a^0.
But a^3/a^3 is just a number divided by itself, which is 1. So a^0 = 1 for any nonzero a.
Negative, Fractional, Decimal, and the Special Exponents
- Negative exponents flip the base into a reciprocal: a^(-n) = 1/a^n, so 2^(-3) = 1/8.
- Fractional exponents are roots in disguise: a^{1/2} = √a and a^{m/n} = √[n]{a^m}.
- Decimal exponents are just fractional exponents written differently: 9^{0.5} = 3.
- Zero and one are boundary cases: a^0 = 1 and a^1 = a.
Examples of Exponents
Example 1
Evaluate 2^5.
Multiply 2 by itself five times.
Final answer: 32.
Example 2
Simplify a^7⋅a^4. When the bases match and you multiply, you add the exponents.
Final answer: a^{11}.
Example 3
Simplify x^9/x^2.
Same base, division, so subtract the exponents.
Final answer: x^7.
Example 4
Evaluate 5^(-2).
A negative exponent means take the reciprocal of the positive power.
Final answer: 1/25.
Example 5
Evaluate 27^{2/3}.
The denominator of the fraction is the root; the numerator is the power. Take the cube root first, then square.
Final answer: 9.
Example 6
Simplify (2x^2 y)^3/4x^3. Distribute the outer power across the product, then simplify. Final answer: 2x^3y^3.
Where Exponents Show Up in the Real World
Exponents describe anything that grows or shrinks by repeated multiplication:
- Compound interest: Money grows by (1+r)^t.
- Computing and storage: Memory sizes are powers of 2.
- Scientific notation: Astronomers use powers of 10.
- Population and decay: Both can be modeled exponentially.
Common Mistakes with Exponents
- Multiplying exponents when you should add them: Don’t multiply when bases match; add instead.
- Mishandling the negative sign: Watch for brackets.
- Adding unlike powers: They cannot combine as terms.
Key Takeaways
- An exponent counts how many times a base is multiplied: b^n = b × b × ... × b (n times).
- The seven laws of exponents all follow from counting factors.
- Exponents model significant real-world phenomena.