Simplify each expression Use only positive exponents. (3x^2y^3)^2 ((2x^4)/(3))^3 (4xy)^-1 (2x^4y^-4z^-3)/(3x^2)y^(-3z^4) (x^2y)/(4)cdot (16x)/(y) (s^4t)^2(st)

## Step 1: Simplify $(3x^{2}y^{3})^{2}$<br />### Apply the power rule $(ab)^n = a^n \cdot b^n$ and distribute the exponent $2$ to each term inside the parentheses. This gives:<br />\[<br />(3x^{2}y^{3})^{2} = 3^{2} \cdot (x^{2})^{2} \cdot (y^{3})^{2}<br />\]<br />Simplify each term:<br />\[<br />3^{2} = 9, \quad (x^{2})^{2} = x^{4}, \quad (y^{3})^{2} = y^{6}<br />\]<br />Thus:<br />\[<br />(3x^{2}y^{3})^{2} = 9x^{4}y^{6}<br />\]<br /><br />## Step 2: Simplify $\left(\frac{2x^{4}}{3}\right)^{3}$<br />### Apply the power rule $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$ and distribute the exponent $3$ to both numerator and denominator:<br />\[<br />\left(\frac{2x^{4}}{3}\right)^{3} = \frac{(2x^{4})^{3}}{3^{3}}<br />\]<br />Simplify each term:<br />\[<br />(2x^{4})^{3} = 2^{3} \cdot (x^{4})^{3} = 8x^{12}, \quad 3^{3} = 27<br />\]<br />Thus:<br />\[<br />\left(\frac{2x^{4}}{3}\right)^{3} = \frac{8x^{12}}{27}<br />\]<br /><br />## Step 3: Simplify $(4xy)^{-1}$<br />### Apply the negative exponent rule $(ab)^{-n} = \frac{1}{(ab)^n}$:<br />\[<br />(4xy)^{-1} = \frac{1}{4xy}<br />\]<br /><br />## Step 4: Simplify $\frac{2x^{4}y^{-4}z^{-3}}{3x^{2}y^{-3}z^{4}}$<br />### Simplify by dividing powers of like bases using the rule $\frac{a^m}{a^n} = a^{m-n}$:<br />\[<br />\frac{2x^{4}y^{-4}z^{-3}}{3x^{2}y^{-3}z^{4}} = \frac{2}{3} \cdot x^{4-2} \cdot y^{-4-(-3)} \cdot z^{-3-4}<br />\]<br />Simplify exponents:<br />\[<br />x^{4-2} = x^{2}, \quad y^{-4-(-3)} = y^{-1}, \quad z^{-3-4} = z^{-7}<br />\]<br />Thus:<br />\[<br />\frac{2x^{4}y^{-4}z^{-3}}{3x^{2}y^{-3}z^{4}} = \frac{2x^{2}}{3y z^{7}}<br />\]<br /><br />## Step 5: Simplify $\frac{x^{2}y}{4} \cdot \frac{16x}{y}$<br />### Multiply numerators and denominators directly:<br />\[<br />\frac{x^{2}y}{4} \cdot \frac{16x}{y} = \frac{x^{2}y \cdot 16x}{4 \cdot y}<br />\]<br />Cancel $y$ in the numerator and denominator:<br />\[<br />\frac{x^{2}y \cdot 16x}{4 \cdot y} = \frac{16x^{3}}{4}<br />\]<br />Simplify:<br />\[<br />\frac{16x^{3}}{4} = 4x^{3}<br />\]<br /><br />## Step 6: Simplify $(s^{4}t)^{2}(st)$<br />### First, simplify $(s^{4}t)^{2}$ using the power rule:<br />\[<br />(s^{4}t)^{2} = (s^{4})^{2} \cdot t^{2} = s^{8}t^{2}<br />\]<br />Now multiply by $(st)$:<br />\[<br />(s^{8}t^{2})(st) = s^{8+1} \cdot t^{2+1} = s^{9}t^{3}<br />\]