Question

select the equivalent expression.
\\(\frac{x^{-2}}{x^{3} \cdot x^{-2}}\\)
answer
\\(\circ\\) \\(x^{3}\\) \\(\circ\\) \\(x^{4}\\) \\(\circ\\) \\(\frac{1}{x^{3}}\\) \\(\circ\\) \\(\frac{1}{x^{4}}\\)

Explanation:

Step1: Simplify the denominator using exponent rule \(a^m \cdot a^n = a^{m + n}\)

The denominator is \(x^3 \cdot x^{-2}\), so applying the rule, we get \(x^{3 + (-2)} = x^{1}\) or \(x\).

Step2: Simplify the fraction using exponent rule \(\frac{a^m}{a^n} = a^{m - n}\)

The expression becomes \(\frac{x^{-2}}{x}\), which is \(x^{-2 - 1}=x^{-3}\).

Step3: Convert negative exponent to positive using \(a^{-n}=\frac{1}{a^n}\)

\(x^{-3}=\frac{1}{x^3}\).

Answer:

\(\frac{1}{x^3}\) (the option corresponding to \(\frac{1}{x^3}\))