Question

Rationalize the denominator of..

Answer 1

\[\left(\begin{matrix}4 \\ 5+\sqrt{2}\end{matrix}\right)\]

Answer 2

Powers that be!

You rationalize a denominator by multiplying by its conjugate, which means you flip the sign before the radical in the denominator and multiply the numerator and the denominator both by that number.

In this case, the denominator is \(5 + \sqrt{2}\), so the conjugate is \(5 - \sqrt{2}\).

Answer 3

You multiply:

\[\frac{4}{5 + \sqrt{2}}\left(\frac{5 - \sqrt{2}}{5 - \sqrt{2}}\right)\]

Answer 4

And you get:

\[\frac{4\cdot 5 - 4\sqrt{2}}{25 + 5\sqrt{2} - 5\sqrt{2} - \sqrt{2}\sqrt{2}}\]

Combine like terms, etc:

\[\frac{20 - 4\sqrt{2}}{25 - 2} = \frac{20 - 4\sqrt{2}}{23}\]

Answer 5

okay thankyou