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
Step 1: Evaluate 9 + 12 ÷ 4 - 2 × 5. Follow the order of operations (PEMDAS/BODMAS): Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), Addition and Subtraction (from left to right). First, perform division: 12 ÷ 4 = 3 Next, perform multiplication: 2 × 5 = 10 Substitute these values back into the expression: 9 + 3 - 10 Now, perform addition and subtraction from left to right: 9 + 3 = 12 12 - 10 = 2 The evaluated expression is: 2 Step 2: Express 125 as a product of its primes. a) To express 125 as a product of its primes, we find the prime factors of 125. 125 is divisible by 5: 125 = 5 × 25 25 is also divisible by 5: 25 = 5 × 5 So, 125 can be written as: 125 = 5 × 5 × 5 = 5^3 The product of its primes is: 5 × 5 × 5 Step 3: Find the cube root of 125. b) Since 125 = 5 × 5 × 5 = 5^3, the cube root of 125 is the number that, when multiplied by itself three times, equals 125. [3]125 = [3]5^3 = 5 The cube root of 125 is: 5 Step 4: Find the L.C.M of 18 and 24. c) To find the Least Common Multiple (L.C.M) of 18 and 24, we first find their prime factorizations. Prime factorization of 18: 18 = 2 × 9 = 2 × 3 × 3 = 2 × 3^2 Prime factorization of 24: 24 = 2 × 12 = 2 × 2 × 6 = 2 × 2 × 2 × 3 = 2^3 × 3 To find the L.C.M, we take the highest power of each prime factor present in either number: The highest power of 2 is 2^3. The highest power of 3 is 3^2. L.C.M(18, 24) = 2^3 × 3^2 = 8 × 9 = 72 The L.C.M of 18 and 24 is: 72 Step 5: Draw a truth table for p q. Given two statements p and q, we need to construct a truth table for the compound statement p q. p means "not p". means "and". | p | q | p | p q | | :-- | :-- | :------- | :--------------- | | T | T | F | F | | T | F | F | F | | F | T | T | T | | F | F | T | F | Step 6: Calculate the distance PQ. a) R is 30km East of P, and Q is 40km South of R. This forms a right-angled triangle PQR, with the right angle at R. The length of PR is 30 km. The length of RQ is 40 km. We can use the Pythagorean theorem to find the distance PQ (the hypotenuse): PQ^2 = PR^2 + RQ^2 PQ^2 = (30 km)^2 + (40 km)^2 PQ^2 = 900 km^2 + 1600 km^2 PQ^2 = 2500 km^2 PQ = sqrt(2500 km)^2 PQ = 50 km The distance PQ is: 50 km Step 7: Calculate the angle QPR. b) In the right-angled triangle PQR, we want to find the angle at P, denoted as QPR. We know the opposite side (RQ = 40 km) and the adjacent side (PR = 30 km) to QPR. We can use the tangent function: ( QPR) = OppositeAdjacent = (RQ)/(PR) ( QPR) = 40 km30 km ( QPR) = (4)/(3) To find the angle, we take the inverse tangent: QPR = ((4)/(3)) Using a calculator: QPR ≈ 53.13^ Rounding to one decimal place: QPR ≈ 53.1^ The angle QPR is: 53.1^