1. **Problem Statement:** Simplify the expressions and solve the systems of equations as given in problems 57, 58, 59, and 60. --- ### 57) Simplify each expression: **a)** Simplify $\left( \frac{3x^0}{y^2} \right)^{-2} \left( \frac{y^4}{3x} \right)$ - Step 1: Recall $x^0 = 1$, so rewrite as $\left( \frac{3}{y^2} \right)^{-2} \cdot \frac{y^4}{3x}$ - Step 2: Apply negative exponent: $\left( \frac{3}{y^2} \right)^{-2} = \left( \frac{y^2}{3} \right)^2 = \frac{y^4}{9}$ - Step 3: Multiply: $\frac{y^4}{9} \cdot \frac{y^4}{3x} = \frac{y^{4+4}}{9 \cdot 3 x} = \frac{y^8}{27x}$ **Answer:** $\boxed{\frac{y^8}{27x}}$ **b)** Simplify $\left( \frac{3y^{-2} z^5}{x^3} \right)^{-3} \left( \frac{x}{34y} \right)^5$ - Step 1: Apply negative exponent: $\left( \frac{3y^{-2} z^5}{x^3} \right)^{-3} = \left( \frac{x^3}{3y^{-2} z^5} \right)^3$ - Step 2: Rewrite $y^{-2}$ as $\frac{1}{y^2}$, so denominator is $3 \cdot \frac{1}{y^2} \cdot z^5 = \frac{3 z^5}{y^2}$ - Step 3: So inside parentheses: $\frac{x^3}{\frac{3 z^5}{y^2}} = \frac{x^3 y^2}{3 z^5}$ - Step 4: Raise to power 3: $\left( \frac{x^3 y^2}{3 z^5} \right)^3 = \frac{x^{9} y^{6}}{3^3 z^{15}} = \frac{x^{9} y^{6}}{27 z^{15}}$ - Step 5: Simplify second term: $\left( \frac{x}{34 y} \right)^5 = \frac{x^5}{34^5 y^5}$ - Step 6: Multiply both: $\frac{x^{9} y^{6}}{27 z^{15}} \cdot \frac{x^5}{34^5 y^5} = \frac{x^{9+5} y^{6-5}}{27 \cdot 34^5 z^{15}} = \frac{x^{14} y}{27 \cdot 34^5 z^{15}}$ **Answer:** $\boxed{\frac{x^{14} y}{27 \cdot 34^5 z^{15}}}$ **c)** Simplify $\left( 3 m^{-3} c^{2} y^{-9} \right)^{-4}$ - Step 1: Apply negative exponent: $= 3^{-4} m^{12} c^{-8} y^{36}$ - Step 2: Simplify $3^{-4} = \frac{1}{3^4} = \frac{1}{81}$ **Answer:** $\boxed{\frac{m^{12} y^{36}}{81 c^{8}}}$ **d)** Simplify $\left( 2^{0} x^{2} y^{-3} z \right)^{15}$ - Step 1: $2^0 = 1$, so expression is $\left( x^{2} y^{-3} z \right)^{15}$ - Step 2: Apply power: $x^{30} y^{-45} z^{15}$ **Answer:** $\boxed{\frac{x^{30} z^{15}}{y^{45}}}$ --- ### 58) Solve for the point $(x,y)$: **a)** System: $$2x + 3y = 24$$ $$x = 4y - 10$$ - Step 1: Substitute $x$ into first equation: $$2(4y - 10) + 3y = 24$$ - Step 2: Expand: $$8y - 20 + 3y = 24$$ - Step 3: Combine like terms: $$11y - 20 = 24$$ - Step 4: Add 20 both sides: $$11y = 44$$ - Step 5: Divide both sides by 11: $$y = \cancel{\frac{44}{11}} = 4$$ - Step 6: Find $x$: $$x = 4(4) - 10 = 16 - 10 = 6$$ **Answer:** $\boxed{(6,4)}$ **b)** System: $$y = x + 7$$ $$x + 2y = -16$$ - Step 1: Substitute $y$: $$x + 2(x + 7) = -16$$ - Step 2: Expand: $$x + 2x + 14 = -16$$ - Step 3: Combine terms: $$3x + 14 = -16$$ - Step 4: Subtract 14: $$3x = -30$$ - Step 5: Divide by 3: $$x = -10$$ - Step 6: Find $y$: $$y = -10 + 7 = -3$$ **Answer:** $\boxed{(-10,-3)}$ **c)** System: $$3x + 2y = 5$$ $$x = y$$ - Step 1: Substitute $x$: $$3y + 2y = 5$$ - Step 2: Combine: $$5y = 5$$ - Step 3: Divide: $$y = 1$$ - Step 4: Find $x$: $$x = 1$$ **Answer:** $\boxed{(1,1)}$ **d)** System: $$2x + 3y = -13$$ $$y = x - 6$$ - Step 1: Substitute $y$: $$2x + 3(x - 6) = -13$$ - Step 2: Expand: $$2x + 3x - 18 = -13$$ - Step 3: Combine: $$5x - 18 = -13$$ - Step 4: Add 18: $$5x = 5$$ - Step 5: Divide: $$x = 1$$ - Step 6: Find $y$: $$y = 1 - 6 = -5$$ **Answer:** $\boxed{(1,-5)}$ --- ### 59) Simplify each complex fraction: **a)** Simplify $\frac{\frac{x}{m}}{d} = \frac{x}{m} \cdot \frac{1}{d} = \frac{x}{md}$ **Answer:** $\boxed{\frac{x}{md}}$ **b)** Simplify $\frac{\frac{1}{r}}{\frac{1}{z}} = \frac{1}{r} \cdot \frac{z}{1} = \frac{z}{r}$ **Answer:** $\boxed{\frac{z}{r}}$ **c)** Simplify $\frac{\frac{N}{a}}{\frac{b}{d}} = \frac{N}{a} \cdot \frac{d}{b} = \frac{N d}{a b}$ **Answer:** $\boxed{\frac{N d}{a b}}$ **d)** Simplify $\frac{w}{\frac{1}{w} + c}$ - Step 1: Find common denominator in denominator: $$\frac{1}{w} + c = \frac{1 + c w}{w}$$ - Step 2: So expression is: $$\frac{w}{\frac{1 + c w}{w}} = w \cdot \frac{w}{1 + c w} = \frac{w^2}{1 + c w}$$ **Answer:** $\boxed{\frac{w^2}{1 + c w}}$ --- ### 60) Graph equations in slope-intercept form: **a)** $y - x = 2 \Rightarrow y = x + 2$ **b)** $3x + 2y = 4 \Rightarrow 2y = -3x + 4 \Rightarrow y = -\frac{3}{2}x + 2$ **c)** $2x - 3y = 6 \Rightarrow -3y = -2x + 6 \Rightarrow y = \frac{2}{3}x - 2$ **d)** $-2y + x = 6 \Rightarrow -2y = -x + 6 \Rightarrow y = \frac{x}{2} - 3$ --- All problems solved with detailed steps and final answers.