1. **Problem:** Simplify the expression $a (5x^2)^3$. 2. **Formula:** Use the power of a product rule: $(ab)^n = a^n b^n$ and the power of a power rule: $(x^m)^n = x^{mn}$. 3. **Step-by-step:** - Expand $(5x^2)^3 = 5^3 (x^2)^3 = 125 x^{6}$. - Multiply by $a$: $a imes 125 x^{6} = 125 a x^{6}$. 4. **Answer:** The simplified form is $125 a x^{6}$. --- 5. **Problem:** Simplify $(16x^4)^{1/2}$. 6. **Formula:** Use the rule $(a^m)^{n} = a^{mn}$ and the square root as power $1/2$. 7. **Step-by-step:** - $(16x^4)^{1/2} = 16^{1/2} imes (x^4)^{1/2} = 4 imes x^{2} = 4 x^{2}$. 8. **Answer:** The simplified form is $4 x^{2}$. --- 9. **Problem:** Simplify $(64x^9)^{1/3}$. 10. **Formula:** Use cube root as power $1/3$. 11. **Step-by-step:** - $(64x^9)^{1/3} = 64^{1/3} imes (x^9)^{1/3} = 4 imes x^{3} = 4 x^{3}$. 12. **Answer:** The simplified form is $4 x^{3}$. --- 13. **Problem:** Matin has 7 number cards with positive numbers. Five cards are [9], [13], [4], [10], [7]. The range is 16 and the mode is 9. Find the two missing numbers. 14. **Formula:** Range = max - min; Mode is the most frequent number. 15. **Step-by-step:** - Known numbers: 4, 7, 9, 10, 13. - Range = 16 means max - min = 16. - Min is 4 (lowest known), so max = 4 + 16 = 20. - Mode is 9, so 9 must appear more than once. - Currently, 9 appears once; to be mode, at least one more 9 is needed. - Two missing numbers: one is 9 (to make mode), the other is 20 (to make range). 16. **Answer:** The missing numbers are 9 and 20. --- 17. **Problem:** Find $x$ in $2^{x+1} = 8^2$. 18. **Formula:** Express both sides with the same base. 19. **Step-by-step:** - $8 = 2^3$, so $8^2 = (2^3)^2 = 2^{6}$. - Equation: $2^{x+1} = 2^{6}$. - Equate exponents: $x + 1 = 6$. - Solve: $x = 5$. 20. **Answer:** $x = 5$. --- 21. **Problem:** Find $x$ in $3^{x - 2/5} = rac{1}{81}$. 22. **Formula:** Express right side as power of 3. 23. **Step-by-step:** - $81 = 3^4$, so $1/81 = 3^{-4}$. - Equation: $3^{x - 2/5} = 3^{-4}$. - Equate exponents: $x - rac{2}{5} = -4$. - Solve: $x = -4 + rac{2}{5} = -rac{20}{5} + rac{2}{5} = -rac{18}{5}$. 24. **Answer:** $x = -rac{18}{5}$. --- 25. **Problem:** Solve $\\sqrt{\frac{11}{25}}$. 26. **Formula:** $\\sqrt{\frac{a}{b}} = rac{\sqrt{a}}{\sqrt{b}}$. 27. **Step-by-step:** - $\\sqrt{\frac{11}{25}} = rac{\sqrt{11}}{5}$. 28. **Answer:** $rac{\sqrt{11}}{5}$. --- 29. **Problem:** Simplify $4x^3 y^2 \times 3x^2 y$. 30. **Formula:** Multiply coefficients and add exponents of like bases. 31. **Step-by-step:** - Coefficients: $4 \times 3 = 12$. - $x^{3} \times x^{2} = x^{5}$. - $y^{2} \times y^{1} = y^{3}$. - Result: $12 x^{5} y^{3}$. 32. **Answer:** $12 x^{5} y^{3}$. --- 33. **Problem:** Simplify $(\frac{4}{5} + \frac{y}{7}) \times \frac{3}{2}$. 34. **Formula:** Distribute multiplication over addition. 35. **Step-by-step:** - Multiply each term by $\frac{3}{2}$: - $\frac{4}{5} \times \frac{3}{2} = \frac{12}{10} = \frac{6}{5}$. - $\frac{y}{7} \times \frac{3}{2} = \frac{3y}{14}$. - Result: $\frac{6}{5} + \frac{3y}{14}$. 36. **Answer:** $\frac{6}{5} + \frac{3y}{14}$.