1. **State the problem:** Simplify the expression $$\frac{\sqrt[3]{-125} \times (1 - 0.8)^2 + 0.3}{(\frac{1}{5})^{-1} - \sqrt{2.25}}$$ 2. **Recall formulas and rules:** - Cube root: $\sqrt[3]{a}$ is the number that when cubed gives $a$. - Powers and roots: $a^{-n} = \frac{1}{a^n}$. - Square root: $\sqrt{a}$ is the number that when squared gives $a$. - Order of operations: parentheses, exponents, multiplication/division, addition/subtraction. 3. **Calculate numerator:** - $\sqrt[3]{-125} = -5$ because $(-5)^3 = -125$. - $1 - 0.8 = 0.2$. - $(0.2)^2 = 0.04$. - Multiply: $-5 \times 0.04 = -0.2$. - Add $0.3$: $-0.2 + 0.3 = 0.1$. 4. **Calculate denominator:** - $(\frac{1}{5})^{-1} = 5$ because $a^{-1} = \frac{1}{a}$. - $\sqrt{2.25} = 1.5$ because $1.5^2 = 2.25$. - Subtract: $5 - 1.5 = 3.5$. 5. **Form the fraction:** $$\frac{0.1}{3.5}$$ 6. **Simplify the fraction:** $$\frac{\cancel{0.1}}{\cancel{3.5}} = \frac{1}{35}$$ 7. **Final answer:** $$\frac{1}{35}$$