Laws of Indices

What an index means

An index is shorthand for repeated multiplication. For a positive whole-number index,

$a^n=\underbrace{a\times a\times\cdots\times a}_{n\text{ factors}}.$

Thus $x^4=x\times x\times x\times x$. The letter $x$ is the base and is the (or ). If no index is shown, it is understood to be , so .

The index laws are not arbitrary instructions. They follow from counting repeated factors. They apply to numerical bases and algebraic bases, but the bases must match where powers are being multiplied or divided.

Multiplying powers with the same base

Writing out $a^3a^2$ gives

$a^3a^2=(a\times a\times a)(a\times a)=a^5.$

There are three copies of $a$ followed by another two, so the indices add:

$a^m\times a^n=a^{m+n}.$

For example, $p^7\times p^4=p^{11}$. If several variables occur, treat each base separately:

$3x^4y^2\times5x^3y=15x^7y^3.$

The coefficients multiply to give $15$; the $x$ indices add; and $$ means , so the becomes .

Different bases cannot be merged by this rule. For example, $x^2y^3$ cannot be rewritten as $x^5$ because $x$ and $$ are not the same base.

Dividing powers with the same base

When like powers are divided, matching factors cancel. For non-zero $a$,

$\frac{a^m}{a^n}=a^{m-n}.$

For instance,

$\frac{x^8}{x^3}=\frac{x\times x\times x\times x\times x\times x\times x\times x}{x\times x\times x}=x^5.$

In an algebraic fraction, simplify coefficients and each variable separately:

$\frac{24x^7y^5}{6x^2y}=4x^5y^4.$

The restriction $a\ne0$ matters because division by zero is undefined.

Raising a power to another power

In $(a^m)^n$, the whole power is used as a factor times. Consequently,

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

For example, $(z^3)^4=z^{12}$. Here the indices are multiplied, not added.

An outside index applies to every factor inside the brackets:

$(2m^4n^2)^3=2^3m^{12}n^6=8m^{12}n^6.$

Two related rules are

$(ab)^n=a^nb^n$

and, when $b\ne0$,

$\left(\frac ab\right)^n=\frac{a^n}{b^n}.$

Why the zero index equals one

Consider dividing a non-zero power by itself:

$\frac{a^4}{a^4}=1.$

The division law gives the same expression as $a^{4-4}=a^0$. Therefore

$a^0=1\qquad(a\ne0).$

In particular, $7^0=1$ and $x^0=1$ for $x\ne0$. The result is not zero.

Negative indices

If the lower power in a division is larger than the upper power, subtraction produces a negative index. For example,

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

Cancelling three copies of $x$ instead leaves $1/x^2$, so

$a^{-n}=\frac{1}{a^n}\qquad(a\ne0).$

A negative index means take the reciprocal; it does not mean that the value is negative. Hence

$4^{-2}=\frac1{4^2}=\frac1{16}.$

It is often clearest to give a simplified algebraic answer using positive indices. For example,

$\frac{6x^2}{3x^5}=2x^{-3}=\frac{2}{x^3}.$

Fractional indices

A fractional index represents a root. For suitable real values of $a$,

$a^{1/n}=\sqrt[n]{a}$

and

$a^{m/n}=\left(\sqrt[n]{a}\right)^m=\sqrt[n]{a^m}.$

The denominator names the root and the numerator names the power. For example,

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

A negative fractional index combines a root, a power and a reciprocal:

$16^{-3/4}=\frac{1}{16^{3/4}}=\frac{1}{(\sqrt[4]{16})^3}=\frac18.$

Taking the root first often keeps the arithmetic manageable. With real numbers, an even root requires a non-negative radicand; for example, $(-16)^{1/2}$ is not a real number.

Sums are different from products

Index laws describe products, quotients and powers—not ordinary addition. Expanding the definitions shows why:

$x^2+x^2=2x^2,$

not $x^4$. The two like terms are added by adding their coefficients. Likewise, cannot be collected because the powers differ.

Nor can a power normally be distributed over a sum:

$(a+b)^2\ne a^2+b^2.$

The brackets make $a+b$ the complete base, and expanding gives $a^2$. Always identify whether the expression contains multiplication, division, a power of a power or a sum before selecting a rule.

Combining several laws

Work in a clear order: simplify numerical coefficients, apply any outside power to every factor, then combine matching . For example,

$\frac{(2x^3y)^2\,x^4}{4xy^3}=\frac{4x^6y^2\,x^4}{4xy^3}=x^9y^{-1}=\frac{x^9}{y}.$

Each change preserves the expression's value: the bracket is squared first, the coefficients cancel, the indices of like bases are subtracted, and the final negative index is written as a reciprocal.

Core laws

For matching bases:

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