1. Simplify each expression or ratio as given. **a)** Simplify $8 \times 8 \times 8 \times 8 \times 8 \times 8$. Since multiplication of the same number repeated is exponentiation, this is $8^6$. **b)** Simplify $5 \times 5 \times 5 \times 2 \times 2 \times 2 \times 2 \times 2$. Group the same bases: $5^3 \times 2^5$. **c)** Simplify the ratio $52 : 38$. Divide both terms by their greatest common divisor (GCD) 2: $$52 : 38 = \cancel{2}26 : \cancel{2}19 = 26 : 19$$ **d)** Simplify the ratio $1.25 : 3.75$. Divide both terms by 1.25: $$1.25 : 3.75 = \cancel{1.25}1 : \cancel{1.25}3 = 1 : 3$$ **e)** Simplify $4^2 \times 4^9$. Use the rule $a^m \times a^n = a^{m+n}$: $$4^2 \times 4^9 = 4^{2+9} = 4^{11}$$ **f)** Simplify the fraction $\frac{115}{112}$. 115 and 112 have no common factors other than 1, so fraction is already simplified. **g)** Simplify $(a^7)^4$. Use the power of a power rule: $(a^m)^n = a^{m \times n}$: $$(a^7)^4 = a^{7 \times 4} = a^{28}$$ **h)** Simplify $\frac{3.25}{12.5}$. Convert decimals to fractions or divide numerator and denominator by 0.25: $$\frac{3.25}{12.5} = \frac{\cancel{0.25}13}{\cancel{0.25}50} = \frac{13}{50}$$ **i)** Simplify $\frac{m^3 \times m^7}{m^6}$. Combine numerator powers: $m^{3+7} = m^{10}$. Divide powers: $\frac{m^{10}}{m^6} = m^{10-6} = m^4$. 2. Fill in missing dimensions using scale factors and ratios. **1.** Given scale factor $6$ mm corresponds to unknown meters. Model length = 38.2 mm, actual length = 98 m. Set ratio: $$\frac{6 \text{ mm}}{x \text{ m}} = \frac{38.2 \text{ mm}}{98 \text{ m}}$$ Cross multiply: $$6 \times 98 = 38.2 \times x$$ $$588 = 38.2x$$ Divide both sides by 38.2: $$x = \frac{588}{38.2} \approx 15.4 \text{ m}$$ **2.** Map scale: 3 in = 2.6 mi. Find map length for actual 4.7 mi. Set ratio: $$\frac{3 \text{ in}}{2.6 \text{ mi}} = \frac{x \text{ in}}{4.7 \text{ mi}}$$ Cross multiply: $$3 \times 4.7 = 2.6 \times x$$ $$14.1 = 2.6x$$ Divide both sides by 2.6: $$x = \frac{14.1}{2.6} \approx 5.4 \text{ in}$$ **3.** Scale factor: 2 in = 8 ft. Find model length for actual 248 ft. Set ratio: $$\frac{2 \text{ in}}{8 \text{ ft}} = \frac{x \text{ in}}{248 \text{ ft}}$$ Cross multiply: $$2 \times 248 = 8 \times x$$ $$496 = 8x$$ Divide both sides by 8: $$x = \frac{496}{8} = 62 \text{ in}$$ **4.** Map scale: 3 cm = unknown km. Given map 1 cm corresponds to 3.4 km. Set ratio: $$\frac{3 \text{ cm}}{x \text{ km}} = \frac{1 \text{ cm}}{3.4 \text{ km}}$$ Cross multiply: $$3 \times 3.4 = 1 \times x$$ $$x = 10.2 \text{ km}$$ 3. Evaluate the expressions. **a)** Evaluate $6 \times \left(3 + \left(\frac{7}{4} - \frac{6}{8}\right) \times 4\right)^3$. Calculate inside parentheses: $$\frac{7}{4} - \frac{6}{8} = \frac{7}{4} - \frac{3}{4} = \frac{4}{4} = 1$$ Multiply by 4: $$1 \times 4 = 4$$ Add 3: $$3 + 4 = 7$$ Cube 7: $$7^3 = 343$$ Multiply by 6: $$6 \times 343 = 2058$$ **b)** Evaluate $\left(1 + \frac{1}{2}\right)^2 - \frac{25 \div 5}{2}$. Calculate inside first parentheses: $$1 + \frac{1}{2} = \frac{3}{2}$$ Square it: $$\left(\frac{3}{2}\right)^2 = \frac{9}{4} = 2.25$$ Calculate division: $$25 \div 5 = 5$$ Divide by 2: $$\frac{5}{2} = 2.5$$ Subtract: $$2.25 - 2.5 = -0.25$$ **c)** Evaluate $\frac{6^2 - 4 \times 3}{2^2 - 8}$. Calculate numerator: $$6^2 = 36$$ $$4 \times 3 = 12$$ $$36 - 12 = 24$$ Calculate denominator: $$2^2 = 4$$ $$4 - 8 = -4$$ Divide: $$\frac{24}{-4} = -6$$