Question

express the following fraction in simplest form, only using positive exponents.\\(\frac{-4(r^{4}t^{-2})^{-1}}{4r^{2}t^{4}}\\)

Explanation:

Step1: Simplify the numerator's exponent

Using the power of a product rule \((ab)^n = a^n b^n\) and the power of a power rule \((a^m)^n = a^{mn}\), we have:
\((r^{4}t^{-2})^{-1}=r^{4\times(-1)}t^{-2\times(-1)} = r^{-4}t^{2}\)
So the numerator becomes \(-4r^{-4}t^{2}\)

Step2: Simplify the fraction by dividing coefficients and using exponent rules for same bases

First, divide the coefficients: \(\frac{-4}{4}=-1\)
For the \(r\) terms: Using the rule \(\frac{a^m}{a^n}=a^{m - n}\), we have \(\frac{r^{-4}}{r^{2}}=r^{-4-2}=r^{-6}\)
For the \(t\) terms: \(\frac{t^{2}}{t^{4}}=t^{2 - 4}=t^{-2}\)
So now we have \(-1\times r^{-6}t^{-2}\)

Step3: Convert negative exponents to positive

Using the rule \(a^{-n}=\frac{1}{a^{n}}\), we get:
\(-1\times\frac{1}{r^{6}}\times\frac{1}{t^{2}}=-\frac{1}{r^{6}t^{2}}\)

Answer:

\(-\frac{1}{r^{6}t^{2}}\)