Problem 10 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

simplify the expression. \(\frac{x^{3}+y^{3}}{x^{2}-x y+y^{2}}\)

Step-by-Step Solution

Answer

The simplified expression is \(x + y\).

Step 1: Factorizing the numerator expression

In the numerator, we have a sum of cubes. We can use the sum of cubes formula: \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\). In this case, the expression is \(x^3 + y^3\), so \(a\) is \(x\) and \(b\) is \(y\). Using the sum of cubes formula, we have: \(x^3 + y^3 = (x + y)(x^2 - xy + y^2)\).

Step 2: Cancel out common terms from numerator and denominator

Now, we have the expression as follows: \(\frac{(x + y)(x^2 - xy + y^2)}{x^2 - xy + y^2}\) We can see that there is one common factor between the numerator and the denominator, which is \((x^2 - xy + y^2)\). Canceling out the common factor, we get: \(\frac{(x + y)(x^2 - xy + y^2)}{x^2 - xy + y^2} = x + y\).

Step 3: Final simplified expression

The simplified expression is: \(x + y\).