Exponents and Powers - Powers with Negative Exponents

Definition of Negative Exponents: A negative exponent represents the reciprocal of the base raised to the positive version of that exponent. Visually, you can think of $a^{-m}$ as moving the base $a^m$ from the numerator to the denominator of a fraction, resulting in $\frac{1}{a^m}$.

Multiplicative Inverse Property: Any non-zero number $a$ raised to the power $-m$ is the multiplicative inverse of $a^m$. This is because their product $a^m \times a^{-m}$ equals $a^0$, which is $1$. Visually, these two values represent opposite sides of a scale that balance at $1$.

Negative Exponents with Fractions: When a fraction $\frac{a}{b}$ is raised to a negative power $-n$, it is equivalent to the reciprocal of the fraction raised to the positive power $n$. This can be visualized as 'flipping' the fraction upside down to get $(\frac{b}{a})^n$.

Product Law for Exponents: When multiplying powers with the same base, we add the exponents: $a^m \times a^n = a^{m+n}$. This applies even if exponents are negative. Visually, this is like combining shifts on a number line where a negative exponent moves the value in the opposite direction of a positive one.

Quotient Law for Exponents: When dividing powers with the same base, we subtract the exponent of the divisor from the exponent of the dividend: $a^m \div a^n = a^{m-n}$. Subtracting a negative exponent is equivalent to adding its absolute value, visually similar to how 'minus a minus' becomes a plus.

Power of a Power Rule: To raise a power to another power, multiply the exponents: $(a^m)^n = a^{mn}$. If one exponent is negative, the final product will be negative, visualizing a repeated reciprocal process.

Representing Small Numbers: Negative exponents are essential for scientific notation of very small numbers. For example, $10^{-3}$ represents $0.001$. Visually, the negative exponent indicates the number of places the decimal point moves to the left of the digit $1$.