You're on a roll — Step 1: Simplify the expression $\frac{30x^4}{3x^{-2}}$. We can separate the coefficients and the variables. $$\frac{30x^4}{3x^{-2}} = \left(\frac{30}{3}\right) \times \left(\frac{x^4}{x^{-2}}\right)$$ First, simplify the coefficients: $$\frac{30}{3} = 10$$ Next, simplify the terms with $x$ using the rule $\frac{a^m}{a^n} = a^{m-n}$: $$\frac{x^4}{x^{-2}} = x^{4 - (-2)} = x^{4+2} = x^6$$ Combine the simplified parts: $$10 \times x^6 = 10x^6$$ $$\boxed{10x^6}$$ Step 2: Simplify $\left(\frac{x^5}{2t^3}\right)^{-4}$. Use the rule $\left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^n$: $$\left(\frac{x^5}{2t^3}\right)^{-4} = \left(\frac{2t^3}{x^5}\right)^4$$ Now, apply the power of 4 to each term in the numerator and the denominator using the rule $(ab)^n = a^n b^n$ and $(a^m)^n = a^{mn}$: $$\left(\frac{2t^3}{x^5}\right)^4 = \frac{(2)^4 (t^3)^4}{(x^5)^4}$$ Calculate the powers: $$(2)^4 = 2 \times 2 \times 2 \times 2 = 16$$ $$(t^3)^4 = t^{3 \times 4} = t^{12}$$ $$(x^5)^4 = x^{5 \times 4} = x^{20}$$ Substitute these back into the expression: $$\frac{16t^{12}}{x^{20}}$$ $$\boxed{\frac{16t^{12}}{x^{20}}}$$ Step 3: Simplify the powers: $\frac{x^5 y^4 z^{-4}}{x^{-2} y^2 z^{-1}}$. Apply the rule $\frac{a^m}{a^n} = a^{m-n}$ to each variable separately: For $x$: $$\frac{x^5}{x^{-2}} = x^{5 - (-2)} = x^{5+2} = x^7$$ For $y$: $$\frac{y^4}{y^2} = y^{4-2} = y^2$$ For $z$: $$\frac{z^{-4}}{z^{-1}} = z^{-4 - (-1)} = z^{-4+1} = z^{-3}$$ Combine the simplified terms. Use the rule $a^{-n} = \frac{1}{a^n}$ for $z^{-3}$: $$x^7 y^2 z^{-3} = \frac{x^7 y^2}{z^3}$$ $$\boxed{\frac{x^7 y^2}{z^3}}$$ Send me the next one 📸