Algebra - Indices (Exponents)

Definition of Indices: An index (or exponent) indicates how many times a base number is multiplied by itself. In the expression $a^n$, $a$ is the base and $n$ is the index. Visually, this can be imagined as a string of $n$ factors of $a$ multiplied together: $a \times a \times a \dots$ ($n$ times).

Product Law: When multiplying two powers with the same base, keep the base and add the exponents. This is visually represented by combining two groups of the same base, one of size $m$ and one of size $n$, to form a single group of size $m+n$.

Quotient Law: When dividing two powers with the same base, keep the base and subtract the exponent of the denominator from the exponent of the numerator. Visually, this corresponds to 'canceling out' common factors of $a$ from the top and bottom of a fraction.

Power of a Power Law: To raise a power to another power, multiply the indices. This can be visualized as a rectangular grid or array where you have $n$ rows, each containing $a^m$, resulting in a total of $m \times n$ factors of $a$.

Negative Indices: A negative exponent represents the reciprocal of the base raised to the positive version of that exponent. Visually, $a^{-n}$ signifies that the base $a^n$ belongs on the opposite side of the fraction line (from numerator to denominator or vice versa).

Zero Index Rule: Any non-zero number raised to the power of zero is equal to $1$. This is a mathematical convention derived from the Quotient Law ($a^n \div a^n = a^{n-n} = a^0 = 1$), indicating that the ratio of any number to itself is always unity.

Fractional Indices and Roots: An index in the form of a fraction $\frac{1}{n}$ represents the $n$-th root of the base. Visually, $a^{\frac{1}{2}}$ is the square root (finding a side of a square with area $a$), and $a^{\frac{1}{3}}$ is the cube root (finding the edge of a cube with volume $a$).

Power of a Product and Quotient: When a product or a quotient is raised to a power, the exponent applies to every factor inside the parentheses. Visually, $(ab)^n$ means $n$ sets of $(a \times b)$, which can be rearranged into $n$ factors of $a$ and $n$ factors of $b$ ($a^n b^n$).