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.
ScanSolve AI Answer
Hey rose, good to see you again. Here are the solutions to your questions: Step 1: Solve Question 11 a) To multiply terms with the same base, add their exponents. a^x × a^x = a^x+x = a^2x The answer is a^2x. b) Calculate the values of each power and then multiply. 5^3 × 3^2 = (5 × 5 × 5) × (3 × 3) 5^3 × 3^2 = 125 × 9 5^3 × 3^2 = 1125 The answer is 1125. c) To multiply terms with the same base, add their exponents. c^x × c^x = c^x+x = c^2x Given c^2x = c^4, the exponents must be equal. 2x = 4 x = (4)/(2) x = 2 The answer is x=2. d) Calculate the values of each power and then subtract. 2^3 - 2^2 = (2 × 2 × 2) - (2 × 2) 2^3 - 2^2 = 8 - 4 2^3 - 2^2 = 4 The answer is 4. e) To solve for x, express 100 as a power of 10. 10^x = 100 10^x = 10^2 Since the bases are the same, the exponents must be equal. x = 2 The answer is x=2. f) The equation is 6^x - 6^x = 216. 0 = 216 This statement is false, so there is no solution for x. The answer is No solution. g) To solve for x, express 16 as a power of 4. 4^x = 16 4^x = 4^2 Since the bases are the same, the exponents must be equal. x = 2 The answer is x=2. Step 2: Solve Question 12 a) Decrease 1000 by 10% then by 20%. First decrease by 10%: 1000 × (1 - 0.10) = 1000 × 0.90 = 900 Then decrease the new amount (900) by 20%: 900 × (1 - 0.20) = 900 × 0.80 = 720 The answer is 720. b) After increasing by 15%, it became 34500. What is the original amount? Let the original amount be A. A × (1 + 0.15) = 34500 A × 1.15 = 34500 A = (34500)/(1.15) A = 30000 The answer is 30000. c) What amount when decreased by 12% becomes 528000? Let the original amount be A. A × (1 - 0.12) = 528000 A × 0.88 = 528000 A = (528000)/(0.88) A = 600000 The answer is 600000. d) When 2000 is decreased by P%, it becomes 16000. Find the value of P. Assumption: The question likely meant "increased by P%" or the initial value was 20000, as a decrease from 2000 cannot result in 16000. I will proceed assuming it meant "When 2000 is increased by P%, it becomes 16000." Let the original amount be 2000. 2000 × (1 + (P)/(100)) = 16000 1 + (P)/(100) = (16000)/(2000) 1 + (P)/(100) = 8 (P)/(100) = 8 - 1 (P)/(100) = 7 P = 7 × 100 P = 700 The answer is P=700. e) Increase 48000 by 12.5% then by 2.25%. First increase by 12.5%: 48000 × (1 + 0.125) = 48000 × 1.125 = 54000 Then increase the new amount (54000) by 2.25%: 54000 × (1 + 0.0225) = 54000 × 1.0225 = 55215 The answer is 55215. Step 3: Solve Question 13 a) Jin covered a distance of 50 km in 2 hours. Calculate speed in km/hr. Speed is calculated as distance divided by time. Speed = DistanceTime Speed = 50 km2 hr Speed = 25 km/hr The speed in km/hr is 25 km/hr. b) What was the speed in m/sec? Convert 25 km/hr to m/sec. 1 km = 1000 m 1 hr = 3600 sec Speed = 25 kmhr × 1000 m1 km × 1 hr3600 sec Speed = (25 × 1000)/(3600) m/sec Speed = (25000)/(3600) m/sec Speed = (250)/(36) m/sec Speed = (125)/(18) m/sec Speed ≈ 6.94 m/sec The speed in m/sec is (125)/(18) m/sec or approximately 6.94 m/sec. Step 4: Solve Question 14 A cyclist covered 200 m in 10 seconds. Find the constant speed in km/hr. First, calculate the speed in m/sec. Speed = DistanceTime Speed = 200 m10 sec Speed = 20 m/sec Now, convert 20 m/sec to km/hr. 1 m = (1)/(1000) km 1 sec = (1)/(3600) hr Speed = 20 msec × 1 km1000 m × 3600 sec1 hr Speed = (20 × 3600)/(1000) km/hr Speed = (72000)/(1000) km/hr Speed = 72 km/hr The constant speed in km/hr is 72 km/hr. Send me the next one 📸