\( \left. \begin{array} { l } { \frac { 12 x ^ { 2 } y ^ { 3 } z ^ { 4 } } { 8 x ^ { 3 } y ^ { 2 } z ^ { 2 } } \times \frac { 8 x ^ { 2 } y ^ { 3 } } { 16 x y } } \\ { \frac { x ^ { 2 } y ^ { 3 } } { ( x ^ { 0 } z ) ^ { 3 } } = \frac { x ^ { - 4 } y ^ { 6 } } { x ^ { 3 } y ^ { - 7 } } } \\ { \frac { 5 x ^ { 3 } \times ( 2 x ) ^ { 2 } } { 20 x ^ { 4 } } } \end{array} \right. \)
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To simplify the first expression, we can start with the fractions. When we divide the polynomials, we subtract the exponents of each variable: \[ \frac{12}{8} = \frac{3}{2}, \quad x^{2-3} = x^{-1}, \quad y^{3-2} = y^{1}, \quad z^{4-2} = z^{2} \] So the first fraction simplifies to \[ \frac{3}{2} \cdot \frac{8}{16} \cdot x^{-1} \cdot y^{1} \cdot z^{2} = \frac{3}{2} \cdot \frac{1}{2} \cdot x^{-1} \cdot y^{1} \cdot z^{2} = \frac{3}{4} y z^{2} x^{-1} \] The second expression can be simplified by noticing \( (x^0 z)^3 = z^3 \), leading to \[ \frac{x^2 y^3}{z^3} = x^{-4} y^{6} / (x^3 y^{-7} ) \] The left-hand side simplifies to \[ x^{2-0} y^{3-0} z^{-3} \text{ and } x^{-4} y^{6+7} z^{3-3} \] If we consider common mistakes, multiplying and dividing variables with exponents is trickier than it seems. Watch out for negative exponents making things a bit confusing. If you haven't practiced enough with linear equations and rational expressions, simplifying an expression correctly can sometimes become a bewildering puzzle! When diving into further reading, consider exploring resources that focus on polynomial long division and the intricacies of rational expressions. Books or online platforms dedicated to algebra can give you a stronger grasp on these foundational concepts, especially if you tackle the exercises that keep your skills sharp and fun!
