Question

express the following fraction in simplest form, only using positive exponents. \\(\dfrac{5(t^{-4}x^{-5})^{-1}}{3t^{4}x^{-1}}\\)

Explanation:

Step1: Simplify the numerator's exponent

First, we deal with the exponent \(-1\) on the numerator \(5(t^{-4}x^{-5})^{-1}\). 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:
\(5(t^{-4}x^{-5})^{-1}=5\times t^{(-4)\times(-1)}\times x^{(-5)\times(-1)} = 5t^{4}x^{5}\)

Step2: Simplify the fraction

Now our fraction is \(\frac{5t^{4}x^{5}}{3t^{4}x^{-1}}\). Using the quotient rule for exponents \(\frac{a^m}{a^n}=a^{m - n}\) (for non - zero \(a\)):
For the \(t\) terms: \(\frac{t^{4}}{t^{4}}=t^{4 - 4}=t^{0}=1\) (since \(t
eq0\))
For the \(x\) terms: \(\frac{x^{5}}{x^{-1}}=x^{5-(-1)}=x^{6}\)
For the constant and the remaining terms, we have \(\frac{5}{3}\times1\times x^{6}=\frac{5x^{6}}{3}\)

Answer:

\(\frac{5x^{6}}{3}\)