Number - Exponents and Powers

An exponent (or power) indicates how many times a base number is multiplied by itself. In the expression $a^n$, $a$ is the base and $n$ is the exponent. Visually, the exponent $n$ is written as a small superscript to the top-right of the base $a$, indicating a 'repeated multiplication' structure.

The Product Law states that when multiplying powers with the same base, you add the indices: $a^m \times a^n = a^{m+n}$. Imagine a row of $m$ factors of $a$ followed by $n$ factors of $a$; when combined, they form a single sequence of $m+n$ factors.

The Quotient Law states that when dividing powers with the same base, you subtract the exponent of the denominator from the exponent of the numerator: $\frac{a^m}{a^n} = a^{m-n}$. Visually, this represents 'canceling out' identical factors from the top and bottom of a fraction until only the difference remains.

The Power of a Power Law states that to raise a power to another power, you multiply the indices: $(a^m)^n = a^{m \times n}$. This can be visualized as a rectangular grid of factors where $a^m$ is repeated $n$ times, resulting in a total count of $m \times n$ bases.

Negative exponents represent the reciprocal of the base raised to the positive power: $a^{-n} = \frac{1}{a^n}$. In a fraction, moving a term from the numerator to the denominator (or vice versa) flips the sign of its exponent, allowing us to represent very small values as fractions.

The Zero Exponent Rule states that any non-zero base raised to the power of zero is equal to $1$ ($a^0 = 1$). This is a logical consequence of the Quotient Law; for example, $\frac{a^n}{a^n} = a^{n-n} = a^0$, and since any number divided by itself is $1$, $a^0$ must be $1$.

Standard Form (Scientific Notation) is a way to write very large or very small numbers in the form $a \times 10^k$, where $1 \le a < 10$ and $k$ is an integer. To convert a decimal, visualize the decimal point 'jumping' left or right; each jump corresponds to a power of $10$.

Fractional exponents relate powers to roots. Specifically, $a^{\frac{1}{n}}$ is equivalent to the $n$-th root of $a$, written as $\sqrt[n]{a}$. For example, $a^{\frac{1}{2}}$ is the square root of $a$, visually represented by the radical symbol $\sqrt{a}$.