Question 275432
{{{5/(3-sqrt(2))}}}
Your denominator has two terms. To rationalize a denominator of two terms we take advantage of the pattern: {{{(a+b)(a-b) = a^2 - b^2}}}. Every two-term expression (aka binomial) will match one of the two factors on the left side of the pattern. In your case, because you have a "-" between the two terms of your denominator, your denominator matches the (a-b) factor in the pattern. If we multiply your denominator by {{{3+sqrt(2)}}} we will get, as we can see from the pattern, two perfect squares (with a "-" between them). This is exactly what you need to rationalize the denominator.<br>
Of course we cannot multiply just the denominator by {{{3+sqrt(2)}}}. We must also multiply the numerator by the same thing. Otherwise we are changing the value of the expression:
{{{(5/(3-sqrt(2)))((3+sqrt(2))/(3+sqrt(2)))}}}
On top we use the Distributive Property. On the bottom, the pattern tells us what we will get:
{{{(15+5sqrt(2))/(3^2-(sqrt(2))^2)}}}
which simplifies as follows:
{{{(15+5sqrt(2))/(9 - 2)}}}
{{{(15+5sqrt(2))/7}}}
and the denominator is now rational.