Question

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

Explanation:

Step1: Apply exponent rule \((a^m)^n = a^{mn}\)

For the numerator's \((t^{-4}x^{-5})^{-1}\), we get \(t^{(-4)\times(-1)}x^{(-5)\times(-1)} = t^{4}x^{5}\). So the numerator becomes \(5t^{4}x^{5}\).

Step2: Rewrite the fraction with new numerator

The fraction is now \(\frac{5t^{4}x^{5}}{3t^{4}x^{-1}}\).

Step3: Apply exponent rule \(\frac{a^m}{a^n}=a^{m - n}\) for \(t\) and \(x\)

For \(t\): \(\frac{t^{4}}{t^{4}}=t^{4 - 4}=t^{0}=1\) (since \(a^{0}=1\) for \(a
eq0\)). For \(x\): \(\frac{x^{5}}{x^{-1}}=x^{5-(-1)} = x^{6}\).

Step4: Simplify the fraction

After simplifying \(t\) terms, we have \(\frac{5\times1\times x^{6}}{3}=\frac{5x^{6}}{3}\).

Answer:

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