Question

express the product $(sqrt{3} - sqrt{5})(sqrt{3} + sqrt{5})$ in simplest form.

Explanation:

Step1: Identify the formula

Notice that \((a - b)(a + b)\) is a difference of squares, which follows the formula \((a - b)(a + b)=a^{2}-b^{2}\). Here, \(a = \sqrt{3}\) and \(b=\sqrt{5}\).

Step2: Apply the formula

Substitute \(a = \sqrt{3}\) and \(b = \sqrt{5}\) into the formula:
\((\sqrt{3})^{2}-(\sqrt{5})^{2}\)

Step3: Simplify the squares

We know that \((\sqrt{x})^{2}=x\) for \(x\geq0\). So, \((\sqrt{3})^{2}=3\) and \((\sqrt{5})^{2}=5\). Then the expression becomes \(3 - 5\).

Step4: Calculate the result

\(3-5=-2\)

Answer:

\(-2\)