Number System - Exponents and Powers: Laws of Exponents, Scientific Notation

Understanding Base and Exponent: An exponent tells how many times a base is multiplied by itself. In the expression $a^n$, $a$ is the base (the number being multiplied) and $n$ is the exponent or index (the number of times the base is used as a factor). Visually, the exponent is written as a small superscript to the top right of the base.

The Product Law: When multiplying two powers with the same base, we keep the base and add the exponents. For example, $2^3 \times 2^4$ means multiplying $2$ three times and then four more times, resulting in $2^{3+4} = 2^7$. This rule only applies when the bases are identical.

The Quotient Law: When dividing two powers with the same base, we keep the base and subtract the exponent of the divisor from the exponent of the dividend. Visually, if we have $\frac{a^m}{a^n}$, we are 'canceling out' $n$ factors of $a$ from the $m$ factors in the numerator, leaving $a^{m-n}$.

Power of a Power Law: When a power is raised to another power, we multiply the exponents. In the expression $(a^m)^n$, the base $a^m$ is being multiplied by itself $n$ times. This simplifies to $a^{m \times n}$.

Zero and Negative Exponents: Any non-zero number raised to the power of zero is always $1$ ($a^0 = 1$). A negative exponent indicates the reciprocal; visually, $a^{-n}$ can be rewritten by moving the base to the denominator of a fraction to make the exponent positive: $\frac{1}{a^n}$.

Scientific Notation (Standard Form): This is used to express very large or very small numbers as the product of a decimal between $1$ and $10$ and a power of $10$. Visually, to convert a number like $4500$ to $4.5 \times 10^3$, the decimal point is moved $3$ places to the left, which becomes the positive exponent of $10$.