Indices — Laws of Indices & Fractional Exponents

Indices are exponents. They show how many times a base is used as a factor, and the laws of indices tell you how to simplify powers without expanding everything. Fractional indices extend the same idea to roots, but the expression still has to be defined.

For a positive integer exponent, $a^n$ means multiply $a$ by itself $n$ times. For example, $2^4 = 2 \cdot 2 \cdot 2 \cdot 2 = 16$.

What the laws of indices say

These are the main rules students use most often:

$$a^m \cdot a^n = a^{m+n}$$ $$\frac{a^m}{a^n} = a^{m-n} \quad (a \ne 0)$$ $$(a^m)^n = a^{mn}$$ $$(ab)^n = a^n b^n$$ $$\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n} \quad (b \ne 0)$$

The conditions matter. You can only add or subtract exponents directly when the base is the same, and quotient rules need a nonzero denominator.

Same base: add when multiplying, subtract when dividing

If the base matches, multiplication combines groups of the same factor:

$$x^3 \cdot x^5 = x^{3+5} = x^8$$

Division removes common factors:

$$\frac{y^7}{y^2} = y^{7-2} = y^5 \quad (y \ne 0)$$

This is the fastest way to avoid a common mistake: $a^m + a^n$ is not the same as $a^{m+n}$. The add-the-exponents rule belongs to multiplication, not addition.

Brackets change the rule

When a power is raised to another power, multiply the exponents:

$$(z^3)^4 = z^{12}$$

When a whole product or quotient is inside brackets, the outside exponent applies to each factor:

$$(2x)^3 = 2^3 x^3 = 8x^3$$ $$\left(\frac{3a}{b}\right)^2 = \frac{9a^2}{b^2} \quad (b \ne 0)$$

Zero, negative, and fractional indices

For any nonzero base,

$$a^0 = 1$$

and

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

A negative exponent does not mean a negative answer. It means take the reciprocal.

Fractional indices connect exponents to roots:

$$a^{1/n} = \sqrt[n]{a}$$ $$a^{m/n} = \left(\sqrt[n]{a}\right)^m$$

For real numbers, the root has to exist. If $n$ is even, you need $a \ge 0$. If $n$ is odd, negative values of $a$ are allowed. So $16^{1/2} = 4$, but $(-16)^{1/2}$ is not a real number.

Worked example: simplify $16^{3/4} \cdot 16^{-1/2}$

Start with the same-base rule:

$$16^{3/4} \cdot 16^{-1/2} = 16^{3/4 - 1/2} = 16^{1/4}$$

Now rewrite the fractional index as a root:

$$16^{1/4} = \sqrt[4]{16} = 2$$

So the full expression simplifies to $2$. This is a good model for many exam questions: combine the exponents first if the base matches, then rewrite the remaining fractional exponent.

Common mistakes with indices

Using the law on addition

$$a^m + a^n \ne a^{m+n}$$

Only multiplication lets you add exponents directly.

Forgetting the same-base condition

$$2^3 \cdot 3^3 = (2 \cdot 3)^3 = 6^3,$$

not $6^6$. The exponents are not added because the original bases were different.

Misreading a negative exponent

$$x^{-2} = \frac{1}{x^2},$$

not $-x^2$.

Ignoring the domain of a fractional exponent

In real-number algebra, $(-9)^{1/2}$ is not real. Before using a root rule, check whether that root exists in the number system you are using.

Where indices are used

Indices show up in algebra, scientific notation, exponential growth and decay, and logarithms. They are useful whenever repeated multiplication, scaling, or powers of $10$ appear.

Try your own version

Try simplifying $x^5 \cdot x^{-2}$, $\frac{a^7}{a^3}$, and $81^{3/4}$. For each one, say which law you used first and check the condition that makes that step valid.