This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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You're on a roll — I can see two questions in the image. I will solve question 2 and question 3. Question 2: The volume of a sphere Step 1: Write down the given formula and values. The volume of a sphere is given by V = (4)/(3) r^3. Given volume V = 38808 cm^3. We need to find the radius r. We will use = (22)/(7). Step 2: Substitute the given values into the formula. 38808 = (4)/(3) × (22)/(7) × r^3 Step 3: Isolate r^3. 38808 = (88)/(21) r^3 Multiply both sides by (21)/(88): r^3 = 38808 × (21)/(88) Step 4: Perform the multiplication and division. First, divide 38808 by 88: 38808 ÷ 88 = 441 Now, multiply by 21: r^3 = 441 × 21 r^3 = 9261 Step 5: Find the cube root of 9261 to get r. r = [3]9261 To find the cube root, we can try small integer values or factorize. We know 20^3 = 8000 and 21^3 = 9261. r = 21 cm The radius of the sphere is 21 cm. --- Question 3: Solve the expression Step 1: Rewrite the expression using fractional exponents. The expression is 5^2 z^1016y^12k^16z^4. This can be written as ((32y^10k^15)^2 z^1016y^12k^16z^4)^(1)/(5). Step 2: Simplify the numerator inside the parenthesis. (32y^10k^15)^2 = 32^2 (y^10)^2 (k^15)^2 = 1024 y^10 × 2 k^15 × 2 = 1024 y^20 k^30 So the numerator becomes 1024 y^20 k^30 z^10. Step 3: Substitute the simplified numerator back into the expression. (1024 y^20 k^30 z^1016y^12k^16z^4)^(1)/(5) Step 4: Simplify the terms inside the parenthesis by dividing coefficients and subtracting exponents for like bases. For the coefficients: (1024)/(16) = 64. For y: y^20y^12 = y^20-12 = y^8. For k: k^30k^16 = k^30-16 = k^14. For z: z^10z^4 = z^10-4 = z^6. So the expression inside the parenthesis simplifies to 64 y^8 k^14 z^6. Step 5: Apply the (1)/(5) exponent to each term. (64 y^8 k^14 z^6)^(1)/(5) = 64^(1)/(5) (y^8)^(1)/(5) (k^14)^(1)/(5) (z^6)^(1)/(5) = 64^(1)/(5) y^(8)/(5) k^(14)/(5) z^(6)/(5) Step 6: Calculate 64^(1)/(5). Since 2^5 = 32 and 3^5 = 243, 64^(1)/(5) is not an integer. We can leave it in this form or approximate it. For exact solution, we leave it as 64^(1)/(5). The simplified expression is 64^(1)/(5) y^(8)/(5) k^(14)/(5) z^(6)/(5). What's next?