Exponents

Exponents record repeated multiplication and provide a compact language for powers, roots, scientific notation, algebra, and growth.

Open the full lesson →

Place value changes by a factor of ten at each step. Adjacent place-value columns are connected by multiply-by-ten and divide-by-ten arrows. A decimal point and scientific-notation form show that notation can change while quantity stays fixed.

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:

a^m a^n=a^{m+n}.

Division cancels matching factors:

\frac{a^m}{a^n}=a^{m-n},\qquad a\ne0.

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:

(a^m)^n=a^{mn}.

An exponent distributes across multiplication and division:

(ab)^n=a^nb^n,\qquad \left(\frac{a}{b}\right)^n=\frac{a^n}{b^n}.

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:

a^{-n}=\frac{1}{a^n}.

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

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?