Exponents

In $a^n$, the base $a$ is multiplied by itself $n$ times when $n$ is a positive whole number.

Meaning and vocabulary

The expression $4^3$ means $4\times4\times4=64$. The base is $4$, the exponent is $3$, and the complete expression is a power. The exponent counts factors—not repeated additions.

Special names are common: $a^2$ is “$a$ squared” and $a^3$ is “$a$ cubed.” Brackets matter. In $(-3)^2$, the base is $-3$ and the value is $9$; in $-3^2$, the exponent applies only to $3$, so the value is $-9$.

Product and quotient rules

When powers have the same base, multiplication combines their factors:

Division cancels matching factors:

These rules do not apply when the bases differ. For example, $2^3\cdot3^3$ may become $(2\cdot3)^3=6^3$, but $2^3\cdot3^4$ cannot be combined by adding exponents.

Power rules

A power raised to another power multiplies the exponents:

An exponent distributes across multiplication and division:

It does not distribute across addition: $(a+b)^2$ is generally not $a^2+b^2$.

Zero and negative exponents

For any nonzero base, $a^0=1$. This follows from $a^3/a^3=a^{3-3}=a^0$, while any nonzero number divided by itself is $1$.

A negative exponent means reciprocal:

The result is not automatically negative. For example, $2^{-3}=1/8$.

Worked simplification

Keep numerical coefficients separate from variable powers. That makes each rule visible and reduces accidental exponent changes.

Exponents and roots

Squaring and taking the principal square root undo one another for nonnegative values:

The absolute value matters because both $5^2$ and $(-5)^2$ equal $25$, while the principal square root $\sqrt{25}$ is $5$.

Order of operations

Evaluate exponents before multiplication, division, addition, and subtraction unless brackets change the order. In $2+3^2(4)$, calculate $3^2=9$, then $9(4)=36$, then $2+36=38$.

Common mistakes

Multiplying the base by the exponent. $5^3$ is $5\cdot5\cdot5$, not $5\cdot3$.

Adding exponents during addition. $x^2+x^3$ cannot become $x^5$; the product $x^2x^3$ can.

Losing brackets around a negative base. $(-2)^4=16$, while $-2^4=-16$.

Making a negative exponent a negative number. Rewrite with a reciprocal instead.

Quick self-check

• Can I name the base and exponent?
• Are the bases identical before I combine powers?
• Does the expression involve multiplication, division, or a power of a power?
• Have I handled brackets and negative signs before calculating?

Related topics

• Order of Operations
• Square Roots
• Scientific Notation
• Polynomial