Problem 39 - College Algebra

Question

Find each product. $$\left(1-y^{5}\right)\left(1+y^{5}\right)$$

Step-by-Step Solution

Answer

The result of the given exercise is \(1 - y^{10}\).

Step 1: Identify the formula

Identify the formula to use to simplify the multiplication. In this case, we use the difference of squares formula which is \(a^2 - b^2 = (a-b)(a + b)\).

Step 2: Apply the formula

Apply the difference of squares formula to the given expressions. For this exercise, these are \(1 - y^{5}\) and \(1 + y^{5}\) Which are in the form \(a-b\) and \(a+b\) respectively. If we substitute these expressions into the formula, we have \((a - b)(a + b) = a^2 - b^2\), which becomes \((1 - y^{5})(1 + y^{5}) = 1^2 - (y^{5})^2\).

Step 3: Finish the calculation

Evaluate \(1^2\) and \((y^{5})^2\), then subtract the square of \(y^{5}\) from the square of 1, we have \(1 - y^{10}\).