## Adding, Subtracting, Dividing or Multiplying Irrational Numbers to Obtain Rational Numbers

\[2+ \sqrt{5}\]

and \[2- \sqrt{5}\]

add to give a rational number.\[(2 + \sqrt{5}) + (2 - \sqrt{5}) = 4\]

Note that the

\[{} + \sqrt{5}\]

cancels the \[- \sqrt{5}\]

.\[ \sqrt{5}\]

and \[2 \sqrt{5}\]

multiply to give a rational number.\[ \sqrt{5}+ \times 2 \sqrt{5} =2 \times \sqrt{25} = 2 \times 5 = 10\]

\[ \sqrt{5}\]

and \[2 \sqrt{5}\]

divide to give a rational number.\[ \frac{\sqrt{5}}{ 2 \sqrt{5}} =\frac{1}{2}\]

\[3+ \sqrt{5}\]

andhj \[2 + \sqrt{5}\]

subtract to give a rational number.\[(3 + \sqrt{5}) - (2 + \sqrt{5}) =3 + \sqrt{5} - 2 - \sqrt{5} =1 \]

Note that the

\[{} + \sqrt{5}\]

cancels the \[{}+ \sqrt{5}\]

.