Question

express as a fraction in simplest form with a rational denominator:
\\(\frac{-8}{7 + \sqrt{2}}\\)

Explanation:

Step1: Identify the conjugate

The conjugate of the denominator \(7 + \sqrt{2}\) is \(7 - \sqrt{2}\). We multiply the numerator and the denominator by this conjugate to rationalize the denominator.

$$ \frac{-8}{7 + \sqrt{2}} \times \frac{7 - \sqrt{2}}{7 - \sqrt{2}} $$

Step2: Multiply the numerators and denominators

First, multiply the numerators: \(-8\times(7 - \sqrt{2})=-56 + 8\sqrt{2}\)
Then, multiply the denominators using the difference of squares formula \((a + b)(a - b)=a^2 - b^2\), where \(a = 7\) and \(b=\sqrt{2}\). So, \((7 + \sqrt{2})(7 - \sqrt{2})=7^2-(\sqrt{2})^2 = 49 - 2=47\)
Now we have the fraction:

$$ \frac{-56 + 8\sqrt{2}}{47} $$

We can also write this as \(\frac{-56 + 8\sqrt{2}}{47}\) or factor out a negative sign from the numerator: \(\frac{-(56 - 8\sqrt{2})}{47}=-\frac{56 - 8\sqrt{2}}{47}\)

Answer:

\(\frac{-56 + 8\sqrt{2}}{47}\) (or equivalent forms like \(-\frac{56 - 8\sqrt{2}}{47}\))