# 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
1. **Multiplying exponents when you should add them**: Don’t multiply when bases match; add instead.
2. **Mishandling the negative sign**: Watch for brackets.
3. **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.