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