1Cademy - Deriving the Definition of a Negative Exponent Using $$\frac{x^2}{x^5}$$

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Quotient Property for Exponents

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Deriving the Definition of a Negative Exponent Using $\frac{x^2}{x^5}$

We can logically deduce the definition of a negative exponent by simplifying the expression $\frac{x^2}{x^5}$ in two different ways. First, if we apply the Quotient Property for Exponents and simply subtract the denominator's exponent from the numerator's exponent, we get $\frac{x^2}{x^5} = x^{2-5} = x^{-3}$ . Second, we can simplify the same expression by dividing out common factors. Writing the numerator as $x \cdot x$ and the denominator as $x \cdot x \cdot x \cdot x \cdot x$ , we can cancel two factors of $x$ from both, which leaves three factors of $x$ in the denominator: $\frac{1}{x \cdot x \cdot x} = \frac{1}{x^3}$ . Because both methods simplify the same original expression, their results must be equal. This implies that $x^{-3} = \frac{1}{x^3}$ . This dual-simplification approach naturally leads to the formal definition of a negative exponent, showing that a negative exponent is equivalent to the reciprocal of the base raised to the positive exponent.

Updated 2026-04-29

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OpenStax Intermediate Algebra

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