Exponents and Powers - Laws of Exponents

An exponent represents repeated multiplication of a number by itself. In the notation $a^n$, $a$ is called the 'base' and $n$ is the 'exponent' or 'power'. Visually, imagine the base $a$ sitting on the line and the exponent $n$ as a smaller number floating at the top-right corner, indicating that $a$ should be written out $n$ times in a row and multiplied.

The Law of Product states that when multiplying powers with the same base, you keep the base and add the exponents: $a^m \times a^n = a^{m+n}$. You can visualize this as taking two separate chains of a multiplied number and linking them together to form one long chain of that same number.

The Law of Quotient applies when dividing powers with the same base: $\frac{a^m}{a^n} = a^{m-n}$. Visually, this is like a fraction where $m$ factors of $a$ are in the numerator and $n$ factors are in the denominator; you 'cancel out' the common factors from both the top and bottom, leaving the difference in the count.

The Power of a Power Law, $(a^m)^n = a^{mn}$, means you multiply the exponents. Visualize this as a nested structure or a grid, where a group of $m$ factors is repeated $n$ times, resulting in a total count of $m \times n$ factors.

When different bases are raised to the same exponent, such as $a^m \times b^m$, they can be combined under a single exponent: $(ab)^m$. Similarly, for division, $\frac{a^m}{b^m} = (\frac{a}{b})^m$. This visualizes the grouping of different numbers into pairs before applying the total number of multiplications.

Negative exponents signify the reciprocal of the base. For any non-zero integer $a$, $a^{-n} = \frac{1}{a^n}$. You can visualize the negative sign as a 'flip' command that moves the base from the numerator to the denominator (or vice versa) to make the exponent positive.

The Zero Exponent rule states that any non-zero number raised to the power of zero is exactly $1$ ($a^0 = 1$). This is visually understood as the result of dividing a number by itself, such as $\frac{a^n}{a^n}$, which simplifies to $1$ while the exponent subtraction $n-n$ results in $0$.

Standard Form or Scientific Notation is used to express very large or very small numbers as $k \times 10^n$, where $1 \le k < 10$. Visually, the exponent $n$ represents the number of places the decimal point has shifted from its original position to create a number between $1$ and $10$.