What an exponent means

Key idea

A power like \(a^n\) means the base \(a\) is multiplied by itself \(n\) times (when \(n\ge 1\)): \[ a^n=\underbrace{a\cdot a\cdot a\cdots a}_{n\text{ factors}}. \] Two useful facts: \(a^1=a\) and \(10^n\) is a 1 followed by \(n\) zeros (for \(n\ge 1\)).

Worked example

Try it

Summary

• \(a^n\) means multiply \(a\) by itself \(n\) times (for \(n\ge 1\)).
• \(a^1=a\). Parentheses matter: \((-2)^2\ne -2^2\).

Multiply powers with the same base

Key idea

When you multiply powers with the same base, you add exponents: \[ a^m\cdot a^n=a^{m+n}. \] This works because you are combining factors of the same base.

Worked example

Try it

Summary

• Product of powers (same base): \(a^m\cdot a^n=a^{m+n}\).
• Do not add exponents if the bases are different.

Divide powers with the same base

Key idea

When you divide powers with the same base (and \(a\ne 0\)), you subtract exponents: \[ \frac{a^m}{a^n}=a^{m-n}. \] A key special case is \(m=n\): \[ \frac{a^m}{a^m}=a^{m-m}=a^0=1. \]

Worked example

Try it

Summary

• Quotient of powers (same base, \(a\ne 0\)): \(\dfrac{a^m}{a^n}=a^{m-n}\).
• Special case: \(\dfrac{a^m}{a^m}=a^0=1\).

Power of a power: \((a^m)^n\)

Key idea

When you raise a power to another power, you multiply exponents: \[ (a^m)^n=a^{mn}. \] This is one of the most common exponent rules in algebra and pre-algebra.

Worked example

Try it

Summary

• Power of a power: \((a^m)^n=a^{mn}\).
• Parentheses tell you what the exponent applies to.

Zero exponents and negative exponents

Key idea

For any nonzero base \(a\): \[ a^0=1 \quad\text{and}\quad a^{-n}=\frac{1}{a^n}. \] Negative exponents do not mean “negative numbers” — they mean “reciprocal.”

Worked example

Try it

Summary

• Zero exponent rule (for \(a\ne 0\)): \(a^0=1\).
• Negative exponent rule (for \(a\ne 0\)): \(a^{-n}=\dfrac{1}{a^n}\).

Combine exponent rules and evaluate

Key idea

When an expression has parentheses, start there. Then apply the exponent rules step by step. A useful checklist:

• Are the bases the same (so you can add/subtract exponents)?
• Is there a power raised to a power (multiply exponents)?
• Do you need to rewrite a negative exponent as a reciprocal?

Worked example

Try it

Summary

• Use parentheses first, then apply exponent rules.
• Convert negative exponents to reciprocals for a final simplified answer.

Why exponents and powers matter

Where you use exponents

• Scientific notation: very large/small numbers using powers of 10 (e.g., \(10^{-2}=0.01\)).
• Geometry: area uses squares (\(cm^2\)), volume uses cubes (\(cm^3\)).
• Computing: powers of 2 show up everywhere (binary, memory sizes, \(2^{10}\)).
• Growth and decay: exponential patterns in science and finance.

Worked example: a power of ten

Try it

Fun facts (a little history)

• Notation: Modern exponent notation became standard in algebra as symbolic math developed in Europe; exponents made repeated multiplication compact and readable.
• Big idea: The same exponent rules power advanced topics like exponential functions, logarithms, and scientific notation.
• Everyday connection: Powers show up in unit conversions (\(m^2\), \(cm^3\)) and in technology (powers of 2).

Final recap

• Meaning (for \(n\ge 1\)): \(a^n=\underbrace{a\cdot a\cdots a}_{n\text{ factors}}\) and \(a^1=a\).
• Product rule (same base): \(a^m a^n=a^{m+n}\).
• Quotient rule (same base, \(a\ne 0\)): \(\dfrac{a^m}{a^n}=a^{m-n}\).
• Power of a power: \((a^m)^n=a^{mn}\).
• Zero exponent (for \(a\ne 0\)): \(a^0=1\).
• Negative exponent (for \(a\ne 0\)): \(a^{-n}=\dfrac{1}{a^n}\).