Business Mathematics and Basic Statistics · Ch 3 — Laws of Indices

Meaning of an Index and the Product/Quotient Laws

This WBCHSE Class 12 Commerce Business Mathematics and Basic Statistics chapter builds a toolkit of rules — the laws of indices (also called laws of exponents) — for simplifying expressions where a number is raised to a power, without ever expanding the power out by repeated multiplication.

What is an index?

For a real number $a \neq 0$ and a positive integer $n$ , the expression $a^n$ (read " $a$ to the power $n$ ") means $a$ multiplied by itself $n$ times:

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

Here $a$ is called the base and $n$ is called the index or exponent.

The product law

When two powers of the SAME base are multiplied, ADD the indices:

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

This follows directly from the definition: $a^m\times a^n$ is $a$ multiplied by itself $m$ times, then $n$ more times — $m+n$ times in total.

The quotient law

When two powers of the SAME base are divided, SUBTRACT the indices:

\frac{a^m}{a^n} = a^{m-n}, \qquad a \neq 0

This follows because dividing by $a^n$ cancels $n$ of the $m$ factors of $a$ in the numerator, leaving $m-n$ factors.

Note

The laws only combine powers of the SAME base

$a^m \times b^n$ (different bases $a$ and $b$ ) CANNOT be simplified by adding the indices — the product and quotient laws apply only when the base is identical on both sides. Always check the bases match before applying either law.

The same laws of indices taught here are a standard, well-established part of algebra taught across commerce-mathematics and general mathematics curricula nationally, applied in this WBCHSE Class 12 Commerce syllabus mainly to simplify algebraic and numerical expressions quickly.

Definition 1Index (Exponent)

In $a^n$ , the number $n$ is the index (or exponent); $a$ is the base. $a^n$ means $a$ multiplied by itself $n$ times, for a positive integer $n$ .

Definition 2Product Law

$a^m \times a^n = a^{m+n}$ — when multiplying powers of the SAME base, add the indices.

Definition 3Quotient Law

$\dfrac{a^m}{a^n} = a^{m-n}$ , for $a \neq 0$ — when dividing powers of the SAME base, subtract the indices.

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