Question

question
select the equivalent expression.
\\(\frac{x^{-7}}{x^{-7} \cdot x^4}\\)
answer
\\(\circ\\ x^{10}\\)
\\(\circ\\ \frac{1}{x^4}\\)
\\(\circ\\ x^4\\)
\\(\circ\\ \frac{1}{x^{10}}\\)

Explanation:

Step1: Simplify the denominator

Using the exponent rule \(a^m \cdot a^n = a^{m + n}\), for the denominator \(x^{-7} \cdot x^4\), we have \(x^{-7 + 4}=x^{-3}\). So the expression becomes \(\frac{x^{-7}}{x^{-3}}\).

Step2: Simplify the fraction

Using the exponent rule \(\frac{a^m}{a^n}=a^{m - n}\), for \(\frac{x^{-7}}{x^{-3}}\), we get \(x^{-7-(-3)} = x^{-7 + 3}=x^{-4}\).

Step3: Rewrite the negative exponent

Using the rule \(a^{-n}=\frac{1}{a^n}\), \(x^{-4}=\frac{1}{x^4}\).

Answer:

\(\frac{1}{x^4}\) (corresponding to the option with \(\frac{1}{x^4}\))