Question

simplify \\(\dfrac{8}{\sqrt{6} + 2}\\). the simplified expression is \\(\square\\).

Explanation:

Step1: Rationalize the denominator

To simplify \(\frac{8}{\sqrt{6} + 2}\), we multiply the numerator and denominator by the conjugate of the denominator, which is \(\sqrt{6}-2\).
So we have \(\frac{8(\sqrt{6}-2)}{(\sqrt{6} + 2)(\sqrt{6}-2)}\)

Step2: Simplify the denominator

Using the difference of squares formula \((a + b)(a - b)=a^{2}-b^{2}\), where \(a=\sqrt{6}\) and \(b = 2\).
The denominator becomes \((\sqrt{6})^{2}-2^{2}=6 - 4=2\)

Step3: Simplify the numerator

Multiply out the numerator: \(8(\sqrt{6}-2)=8\sqrt{6}-16\)

Step4: Divide numerator by denominator

Now we have \(\frac{8\sqrt{6}-16}{2}\), we can divide each term in the numerator by 2.
\(\frac{8\sqrt{6}}{2}-\frac{16}{2}=4\sqrt{6}-8\)

Answer:

\(4\sqrt{6}-8\)