Question

select the equivalent expression.
\\(\frac{x^{-4}}{x^{8} \cdot x^{-7}}\\)
answer
\\(\frac{1}{x^5}\\) \\(x^5\\) \\(x^{19}\\) \\(\frac{1}{x^{19}}\\)

Explanation:

Step1: Simplify the denominator using exponent rule

When multiplying exponents with the same base, we add the exponents. So for \(x^{8}\cdot x^{-7}\), we use the rule \(a^{m}\cdot a^{n}=a^{m + n}\).
So \(x^{8}\cdot x^{-7}=x^{8+( - 7)}=x^{1}\)

Step2: Simplify the fraction using exponent rule

Now we have \(\frac{x^{-4}}{x^{1}}\). When dividing exponents with the same base, we subtract the exponents. The rule is \(\frac{a^{m}}{a^{n}}=a^{m - n}\)
So \(\frac{x^{-4}}{x^{1}}=x^{-4-1}=x^{-5}\)

Step3: Convert negative exponent to positive

Recall that \(a^{-n}=\frac{1}{a^{n}}\). So \(x^{-5}=\frac{1}{x^{5}}\)

Answer:

\(\frac{1}{x^{5}}\) (corresponding to the option \(\frac{1}{x^{5}}\))