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
Okay, Empress🥰🥰🥰, let's break down these two problems. Problem 1: Logarithm Calculation Calculate (15) - 2 ((5)/(4)) + ((100)/(243)) Step 1: Apply the power rule of logarithms, n x = x^n. 2 ((5)/(4)) = (((5)/(4))^2) = ((25)/(16)) The expression becomes: (15) - ((25)/(16)) + ((100)/(243)) Step 2: Apply the quotient and product rules of logarithms, x - y = ((x)/(y)) and x + y = (xy). Combine the terms into a single logarithm: ( (15)/(25)16 × (100)/(243) ) ( 15 × (16)/(25) × (100)/(243) ) Step 3: Simplify the expression inside the logarithm. ( (15 × 16 × 100)/(25 × 243) ) Cancel common factors: ( ((3 × 5) × 16 × (4 × 25))/(25 × (3 × 81)) ) Cancel 5 and 25 (from 15/25): ( (3 × 16 × 100)/(5 × 243) ) Cancel 5 and 100: ( (3 × 16 × 20)/(243) ) Cancel 3 and 243: ( (16 × 20)/(81) ) ( (320)/(81) ) The final answer is ((320)/(81)). Problem 2: Scientific Notation Calculation Calculate (0.00042 × 10^-8)(15,000)(5000 × 10^7)(0.0021 × 10^14) Step 1: Convert all numbers to scientific notation. 0.00042 = 4.2 × 10^-4 15,000 = 1.5 × 10^4 5000 = 5 × 10^3 0.0021 = 2.1 × 10^-3 Substitute these into the expression: (4.2 × 10^-4 × 10^-8)(1.5 × 10^4)(5 × 10^3 × 10^7)(2.1 × 10^-3 × 10^14) Step 2: Group the numerical parts and the powers of 10 in the numerator and denominator. Numerator: (4.2 × 1.5) × (10^-4 × 10^-8 × 10^4) = 6.3 × 10^(-4 - 8 + 4) = 6.3 × 10^-8 Denominator: (5 × 2.1) × (10^3 × 10^7 × 10^-3 × 10^14) = 10.5 × 10^(3 + 7 - 3 + 14) = 10.5 × 10^21 The expression becomes: 6.3 × 10^-810.5 × 10^21 Step 3: Divide the numerical parts and the powers of 10. Divide the numerical parts: (6.3)/(10.5) = (63)/(105) Both 63 and 105 are divisible by 21: 63 ÷ 21 = 3 and 105 ÷ 21 = 5. (3)/(5) = 0.6 Divide the powers of 10: 10^-810^21 = 10^-8 - 21 = 10^-29 Combine these results: 0.6 × 10^-29 Step 4: Convert the result to standard scientific notation. 0.6 × 10^-29 = (6 × 10^-1) × 10^-29 = 6 × 10^(-1 - 29) = 6 × 10^-30 The final answer is 6 × 10^-30. Last free one today — make it count tomorrow, or type /upgrade for unlimited.