What does x to the power of -n represent?

When a variable is raised to a negative exponent, it represents the reciprocal of that variable raised to the corresponding positive exponent. Therefore, (x^{-n}) indicates the expression (1) divided by (x) raised to the power of (n). This concept stems from the property of exponents that states (a^{-b} = \frac{1}{a^b}), where (a) is any non-zero number and (b) is a positive integer.

Applying this property, we see that for (x^{-n}), we rewrite it as:

[

x^{-n} = \frac{1}{x^n}

]

Thus, the correct interpretation of (x) raised to the power of (-n) is indeed equivalent to (1) divided by (x) raised to the power of (n). This understanding solidifies the relationship between negative exponents and their positive counterparts, reinforcing the rules of exponents that are fundamental in algebra.