Number Systems - Laws of Exponents for Real Numbers

Definition of Exponents: For any real number $a$ and a positive integer $n$, the expression $a^n$ represents the product of $a$ multiplied by itself $n$ times. Visually, the base $a$ is written as a standard-sized number, while the exponent $n$ is written as a smaller superscript at the top right corner.

Product Law: When multiplying powers with the same base, you keep the base and add the exponents ($a^m \cdot a^n = a^{m+n}$). This visualizes as merging two groups of identical factors into one single group containing all factors.

Quotient Law: When dividing powers with the same base, you keep the base and subtract the exponent of the denominator from the exponent of the numerator ($\frac{a^m}{a^n} = a^{m-n}$). This represents the process of 'canceling out' matching factors from the top and bottom of a fraction.

Power of a Power Law: When a power is raised to another power, the exponents are multiplied together ($(a^m)^n = a^{mn}$). Visually, this represents a block of $a$ factors repeated $n$ times, resulting in a single base with a larger product exponent.

Power of a Product and Quotient: An exponent outside a bracket containing a product or quotient applies to every individual term inside the bracket. For example, $(ab)^n = a^n b^n$ and $(\frac{a}{b})^n = \frac{a^n}{b^n}$. This acts like a distributive property for exponents over multiplication and division.

Zero and Negative Exponents: Any non-zero real number raised to the power of 0 is always 1 ($a^0 = 1$). A negative exponent indicates a reciprocal; $a^{-n}$ moves the base to the denominator with a positive exponent ($a^{-n} = \frac{1}{a^n}$).

Rational Exponents: For a real number $a > 0$ and integers $m$ and $n$, $a^{\frac{m}{n}}$ represents the $n^{th}$ root of $a$ raised to the $m^{th}$ power. Visually, the denominator of the fraction sits inside the notch of the radical sign $\sqrt[n]{a^m}$.