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.
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You're on a roll — here are the solutions to the problems: 1) (a^2)/(a-2b) + (4b^2)/(2b-a) Step 1: Rewrite the denominator of the second term. Note that 2b-a = -(a-2b). (a^2)/(a-2b) + (4b^2)/(-(a-2b)) = (a^2)/(a-2b) - (4b^2)/(a-2b) Step 2: Combine the fractions with the common denominator. (a^2 - 4b^2)/(a-2b) Step 3: Factor the numerator using the difference of squares formula (x^2-y^2 = (x-y)(x+y)). ((a-2b)(a+2b))/(a-2b) Step 4: Cancel out the common factor (a-2b). a+2b The simplified expression is a+2b. 2) (1)/(b-1) - (1)/(b+1) Step 1: Find a common denominator, which is (b-1)(b+1). (1 · (b+1))/((b-1)(b+1)) - (1 · (b-1))/((b+1)(b-1)) Step 2: Perform the subtraction. ((b+1) - (b-1))/((b-1)(b+1)) Step 3: Simplify the numerator and the denominator. (b+1-b+1)/(b^2-1) = (2)/(b^2-1) The simplified expression is (2)/(b^2-1). 3) (1)/(x-1) - (2)/(x^2-1) Step 1: Factor the denominator of the second term using the difference of squares formula. x^2-1 = (x-1)(x+1) (1)/(x-1) - (2)/((x-1)(x+1)) Step 2: Find a common denominator, which is (x-1)(x+1). (1 · (x+1))/((x-1)(x+1)) - (2)/((x-1)(x+1)) Step 3: Perform the subtraction. ((x+1) - 2)/((x-1)(x+1)) Step 4: Simplify the numerator. (x-1)/((x-1)(x+1)) Step 5: Cancel out the common factor (x-1). (1)/(x+1) The simplified expression is (1)/(x+1). 4) (a^3+1)/(a^2-a+1) + (a^3-1)/(a^2+a+1) Step 1: Factor the numerators using the sum and difference of cubes formulas: x^3+y^3 = (x+y)(x^2-xy+y^2) and x^3-y^3 = (x-y)(x^2+xy+y^2). a^3+1 = (a+1)(a^2-a+1) a^3-1 = (a-1)(a^2+a+1) Step 2: Substitute the factored forms into the expression. ((a+1)(a^2-a+1))/(a^2-a+1) + ((a-1)(a^2+a+1))/(a^2+a+1) Step 3: Cancel out common factors in each term. (a+1) + (a-1) Step 4: Simplify the expression. a+1+a-1 = 2a The simplified expression is 2a. 5) (m^2+mn+n^2)/(m+n) + (m^2-mn+n^2)/(m-n) Step 1: Find a common denominator, which is (m+n)(m-n) = m^2-n^2. ((m^2+mn+n^2)(m-n))/((m+n)(m-n)) + ((m^2-mn+n^2)(m+n))/((m-n)(m+n)) Step 2: Expand the numerators. The first numerator is (m^2+mn+n^2)(m-n) = m^3-n^3. The second numerator is (m^2-mn+n^2)(m+n) = m^3+n^3. Step 3: Substitute the expanded numerators back into the expression. (m^3-n^3)/(m^2-n^2) + (m^3+n^3)/(m^2-n^2) Step 4: Combine the fractions. ((m^3-n^3) + (m^3+n^3))/(m^2-n^2) Step 5: Simplify the numerator. (m^3-n^3+m^3+n^3)/(m^2-n^2) = (2m^3)/(m^2-n^2) The simplified expression is (2m^3)/(m^2-n^2). What's next?