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
Here are the values of the given expressions using appropriate identities. i) We need to find the value of (997)^3. We can write 997 as (1000 - 3). We use the identity (a-b)^3 = a^3 - 3a^2b + 3ab^2 - b^3. Here, a = 1000 and b = 3. Step 1: Substitute a and b into the identity. (997)^3 = (1000 - 3)^3 = (1000)^3 - 3(1000)^2(3) + 3(1000)(3)^2 - (3)^3 Step 2: Calculate each term. (1000)^3 = 1,000,000,000 3(1000)^2(3) = 3(1,000,000)(3) = 9,000,000 3(1000)(3)^2 = 3(1000)(9) = 27,000 (3)^3 = 27 Step 3: Combine the terms. = 1,000,000,000 - 9,000,000 + 27,000 - 27 = 991,000,000 + 27,000 - 27 = 991,027,000 - 27 = 991,026,973 The value is: 991,026,973 ii) We need to find the value of (103)^3. We can write 103 as (100 + 3). We use the identity (a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3. Here, a = 100 and b = 3. Step 1: Substitute a and b into the identity. (103)^3 = (100 + 3)^3 = (100)^3 + 3(100)^2(3) + 3(100)(3)^2 + (3)^3 Step 2: Calculate each term. (100)^3 = 1,000,000 3(100)^2(3) = 3(10,000)(3) = 90,000 3(100)(3)^2 = 3(100)(9) = 2,700 (3)^3 = 27 Step 3: Combine the terms. = 1,000,000 + 90,000 + 2,700 + 27 = 1,090,000 + 2,700 + 27 = 1,092,700 + 27 = 1,092,727 The value is: 1,092,727 3 done, 2 left today. You're making progress.