The laws of indices

An index tells you how many times a number or expression is used as a factor. For example,

$x^4=x\times x\times x\times x.$

The familiar index laws are not disconnected rules. They all follow from this idea and from extending it consistently to zero, negative and fractional powers.

Multiplying powers with the same base

When the base is the same, multiplying joins the factors together:

$x^a\times x^b=x^{a+b}.$

For example,

$x^3\times x^5=(x\times x\times x)(x\times x\times x\times x\times x)=x^8.$

The bases must match. You cannot simplify $x^3y^5$ to a single power because $x$ and $y$ may have different values.

Dividing powers with the same base

Division cancels matching factors, so

$\frac{x^a}{x^b}=x^{a-b},\qquad x\ne0.$

If $a<b$, the result has a negative index. For instance,

$\frac{x^2}{x^5}=x^{-3}=\frac{1}{x^3}.$

This explains the negative-index rule rather than requiring a new one.

A power raised to a power

In $(x^a)^b$, the factor $x^a$ is repeated $b$ times. Therefore,

$(x^a)^b=x^{ab}.$

Be careful not to add the indices here. For example, $(x^3)^4=x^{12}$, not $x^7$.

Zero and fractional indices

For non-zero $x$,

$x^0=1.$

You can see this from division: $x^3/x^3=x^{3-3}=x^0$, while any non-zero number divided by itself is $1$.

A fractional index describes a root:

$x^{1/n}=\sqrt[n]{x},\qquad x^{m/n}=\sqrt[n]{x^m}=(\sqrt[n]{x})^m.$

For example,

$27^{2/3}=(\sqrt[3]{27})^2=3^2=9.$

Common mistakes

• The addition rule applies when multiplying, not when adding: $x^2+x^3$ does not equal $x^5$.
• $(x+y)^2$ is not $x^2+y^2$; expanding gives $x^2+2xy+y^2$.
• A negative index creates a reciprocal. It does not make the value negative.
• Apply powers to every factor: $(3x^2)^3=27x^6$.

Quick check

Simplify $\dfrac{(2x^3)^2}{4x^{-1}}$.

The numerator is $4x^6$, so

$\frac{4x^6}{4x^{-1}}=x^{6-(-1)}=x^7.$