Questions: Express the following fraction in simplest form, only using positive exponents. 5(t^(-4) x^(-5))^(-1) / (3 t^4 x^(-1))

Solution Steps

The numerator is \(5\left(t^{-4} x^{-5}\right)^{-1}\). Using the power of a power rule \((a^m)^n = a^{m \cdot n}\), we simplify the expression inside the parentheses: \[ \left(t^{-4} x^{-5}\right)^{-1} = t^{4} x^{5}. \] Thus, the numerator becomes: \[ 5 \cdot t^{4} x^{5}. \]

The denominator is \(3 t^{4} x^{-1}\). Rewrite \(x^{-1}\) as \(\frac{1}{x}\): \[ 3 t^{4} x^{-1} = 3 t^{4} \cdot \frac{1}{x}. \]

Now, divide the numerator by the denominator: \[ \frac{5 t^{4} x^{5}}{3 t^{4} \cdot \frac{1}{x}}. \] Simplify by canceling \(t^{4}\) in the numerator and denominator: \[ \frac{5 x^{5}}{3 \cdot \frac{1}{x}}. \] Multiply by the reciprocal of \(\frac{1}{x}\): \[ \frac{5 x^{5} \cdot x}{3} = \frac{5 x^{6}}{3}. \]

Final Answer