Exponents and Powers - Decimal Number System

Understanding Base and Exponents: In the expression $10^n$, $10$ is the base and $n$ is the exponent or power. This notation represents multiplying the base by itself $n$ times. Visually, you can imagine a chain of multiplication: $10 \times 10 \times 10 \dots$ repeated $n$ times.

The Decimal Number System and Powers of 10: Our number system is based on powers of $10$. Each position in a number represents a specific power of $10$. Starting from the right (units place), the positions represent $10^0$, $10^1$ (tens), $10^2$ (hundreds), $10^3$ (thousands), and so on.

Expanded Form: Any number can be written as a sum of its digits multiplied by their respective place values expressed as powers of $10$. For example, the number $5427$ is visualized as 4 distinct parts added together: $(5 \times 10^3) + (4 \times 10^2) + (2 \times 10^1) + (7 \times 10^0)$.

Standard Form (Scientific Notation): To handle very large numbers easily, we express them in standard form as $k \times 10^n$. Here, $k$ is a decimal number such that $1 \le k < 10$, and $n$ is a positive integer. Visually, this is done by placing a decimal point after the first non-zero digit and counting how many places it moved to determine $n$.

The Zero Exponent Rule: Any non-zero number raised to the power of zero is always $1$. In the decimal system, this explains why the units place digit is multiplied by $10^0$ (which equals $1$). Thus, $a^0 = 1$ for any $a \neq 0$.

Multiplying by Powers of 10: When a decimal is multiplied by $10^n$, the decimal point shifts $n$ places to the right. Conversely, when dividing by $10^n$, the decimal point shifts $n$ places to the left. This visual movement of the point is the foundation of scientific notation conversion.

Laws of Exponents for Calculation: While working with powers of $10$, we use laws like $a^m \times a^n = a^{m+n}$ and $a^m \div a^n = a^{m-n}$. This helps in simplifying expressions where the base ($10$) is the same.