Exponents and Powers - Laws of Exponents with integral powers

Definition of Exponents: An exponent (or power) indicates how many times a base is multiplied by itself. In the expression $a^n$, $a$ is the 'base' and $n$ is the 'exponent'. Visually, imagine $a^3$ as three copies of $a$ lined up horizontally and multiplied together: $a \times a \times a$.

Negative Integral Exponents: A negative exponent represents the reciprocal of the base raised to the positive version of that exponent. If you see $a^{-n}$, you can visually interpret the negative sign as a command to 'flip' the base to the denominator, resulting in $\frac{1}{a^n}$. For example, $3^{-2}$ becomes $\frac{1}{3^2}$ or $\frac{1}{9}$.

The Product Law: When multiplying powers with the same base, you keep the base and add the exponents. This is because you are aggregating the total number of times the base appears in a product chain. Visually, $x^2 \times x^3$ is $(x \cdot x) \times (x \cdot x \cdot x)$, which combines into a single string of five $x$s, or $x^5$.

The Quotient Law: When dividing powers with the same base, you subtract the exponent of the divisor from the exponent of the dividend. In a fractional view, this represents 'canceling out' identical factors from the top and bottom. For instance, $\frac{y^5}{y^2}$ means five $y$s on top and two on the bottom; after removing the pairs, three $y$s remain on top, represented as $y^{5-2} = y^3$.

Power of a Power Law: When a power is raised to another exponent, you multiply the exponents together. Visually, $(a^m)^n$ can be seen as a grid where you have $n$ rows, and each row contains $m$ factors of $a$, leading to a total of $m \times n$ factors. Example: $(2^3)^2 = 2^{3 \times 2} = 2^6$.

Power of a Product and Quotient: An exponent outside a bracket applies to every factor inside the bracket. For products $(ab)^n = a^n b^n$, and for quotients $(\frac{a}{b})^n = \frac{a^n}{b^n}$. Imagine the exponent 'distributing' itself to each term within the parentheses.

Zero Exponent Rule: Any non-zero base raised to the power of zero is equal to $1$. This concept is derived from the quotient law: $\frac{a^n}{a^n} = a^{n-n} = a^0$. Since any number divided by itself is $1$, it follows that $a^0 = 1$. Visually, this represents an empty product, which is defined as the multiplicative identity, $1$.