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 let you simplify powers without expanding everything. For a positive integer exponent, $a^n$ means multiply $a$ by itself $n$ times, so $2^4 = 2 \cdot 2 \cdot 2 \cdot 2 = 16$. Fractional indices extend the same idea to roots, as long as the expression stays defined.

When Each Law Applies

Before simplifying, match the expression to the right law. These are the rules students use most, each with the condition that keeps it valid:

You can add or subtract exponents directly only when the base is the same, and quotient rules need a nonzero denominator. Picking the law by the structure of the expression is the whole game.

Same Base: Add When Multiplying, Subtract When Dividing

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

Division removes common factors:

The add-the-exponents rule belongs to multiplication, not addition: $a^m + a^n$ is not $a^{m+n}$.

Brackets Change The Rule

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

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

Zero, Negative, And Fractional Indices

For any nonzero base,

and

A negative exponent does not mean a negative answer; it means take the reciprocal. Fractional indices connect exponents to roots:

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

The Whole Procedure On One Example

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

The bases match, so combine the exponents with the same-base rule first:

Now rewrite the fractional index as a root:

So the expression simplifies to $2$. That order, combine matching bases first, then convert the leftover fractional exponent to a root, models most exam questions.

Where Each Step Goes Wrong, And How To Check

• Applying the law to addition. $a^m + a^n \ne a^{m+n}$; only multiplication lets you add exponents.
• Forgetting the same-base condition. $2^3 \cdot 3^3 = (2 \cdot 3)^3 = 6^3$, not $6^6$, because the original bases differ.
• 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, so check that the root exists before using a root rule.

A fast self-check for each step: name the law you used, then verify its condition (matching base, nonzero base or denominator, valid root) is actually met.

Where Indices Are Used

Indices show up in algebra, scientific notation, exponential growth and decay, and logarithms, wherever repeated multiplication, scaling, or powers of $10$ appear. For a quick run-through, simplify $x^5 \cdot x^{-2}$, $\frac{a^7}{a^3}$, and $81^{3/4}$, and for each one state which law you applied first and confirm the condition that makes that step valid.