Exponents and Powers - Negative Exponents

Negative Exponent Definition: For any non-zero rational number $a$ and a positive integer $n$, $a^{-n} = \\frac{1}{a^n}$. Visual: Think of the negative sign in the exponent as a 'directional switch'; it indicates that the base belongs on the opposite side of the fraction bar (the denominator) to make the exponent positive.

Reciprocal Rule for Fractions: When a fraction is raised to a negative power, it is equal to the reciprocal of that fraction raised to the same positive power: $(\\frac{a}{b})^{-n} = (\\frac{b}{a})^n$. Visual: Imagine the fraction 'flipping' 180 degrees so the numerator and denominator swap positions as the negative sign in the power disappears.

Multiplicative Inverse Property: The term $a^{-n}$ is the multiplicative inverse of $a^n$ because their product is $a^n \\times a^{-n} = a^0 = 1$. Visual: Just as $\\frac{1}{2}$ is the inverse of $2$, $a^{-n}$ is the 'partner' to $a^n$ that balances the expression to equal $1$.

Product Law with Negative Indices: The rule $a^m \\times a^n = a^{m+n}$ remains valid even when $m$ or $n$ are negative integers. Visual: Multiplying by $a^{-3}$ is equivalent to moving $3$ units to the left on an exponent number line, effectively reducing the total power.

Quotient Law with Negative Indices: To divide powers with the same base, subtract the exponents: $a^m \\div a^n = a^{m-n}$. Visual: Subtracting a negative exponent (e.g., $m - (-n)$) is visually and mathematically equivalent to 'lifting' the base from the denominator into the numerator and adding the power.

Power of a Power Law: $(a^m)^n = a^{mn}$. If one of the exponents is negative, the resulting product will be negative, placing the final result in the denominator. Visual: Think of the outer exponent as a multiplier that scales the inner power, regardless of its sign.

The Zero Base Restriction: A negative exponent cannot be applied to a base of zero ($0^{-n}$ is undefined). Visual: Since $a^{-n}$ involves placing the base in the denominator, a zero base would result in division by zero, which creates a mathematical error or an 'undefined' state.