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 simplified forms of the given expressions using appropriate identities. i) We need to simplify x^3 - 8y^3 - 6x^2y + 12xy^2. This expression resembles the expansion of (a-b)^3 = a^3 - 3a^2b + 3ab^2 - b^3. Step 1: Identify a and b from the cubic terms. The first term is x^3, so a = x. The second term is -8y^3, which can be written as -(2y)^3. So, b = 2y. Step 2: Verify the middle terms using a=x and b=2y. The term -3a^2b = -3(x)^2(2y) = -6x^2y. This matches the third term in the given expression. The term 3ab^2 = 3(x)(2y)^2 = 3(x)(4y^2) = 12xy^2. This matches the fourth term in the given expression. Step 3: Write the expression in its compact form. Since all terms match the expansion of (a-b)^3, we can write the given expression as (x - 2y)^3. (x - 2y)^3 ii) We need to simplify 27a^3 + (1)/(64b^3) + (27a^2)/(4b) + (9a)/(16b^2). This expression resembles the expansion of (A+B)^3 = A^3 + 3A^2B + 3AB^2 + B^3. Step 1: Identify A and B from the cubic terms. The first term is 27a^3, which can be written as (3a)^3. So, A = 3a. The second term is (1)/(64b^3), which can be written as ((1)/(4b))^3. So, B = (1)/(4b). Step 2: Verify the middle terms using A=3a and B=(1)/(4b). The term 3A^2B = 3(3a)^2((1)/(4b)) = 3(9a^2)((1)/(4b)) = (27a^2)/(4b). This matches the third term in the given expression. The term 3AB^2 = 3(3a)((1)/(4b))^2 = 3(3a)((1)/(16b^2)) = (9a)/(16b^2). This matches the fourth term in the given expression. Step 3: Write the expression in its compact form. Since all terms match the expansion of (A+B)^3, we can write the given expression as (3a + (1)/(4b))^3. (3a + (1)/(4b))^3 What's next? Send 'em!