Question 242673
{{{(sqrt(9)+4)/(sqrt(8) + 6)}}}
With a square root (which is irrational) in the denominator, the problem is to rationalize the denominator. But before we start we are going to simplify the square roots we do have.<br>
{{{sqrt(9) = 3}}}
{{{sqrt(8) = sqrt(4*2) = sqrt(4)*sqrt(2) = 2sqrt(2)}}}
Substituting for these two we get:
{{{((3)+4)/(2sqrt(2) + 6)}}}
which simplifies to:
{{{7/(2sqrt(2) + 6)}}}<br>
Now we can try to rationalize the denominator. With two terms, this denominator is a little harder to rationalize than if it was just one term. To rationalize two-term denominators we take advantage of the pattern: {{{a+b)(a-b) = a^2 - b^2}}}. All two-term expressions fit either the (a+b) or the (a-b) part of the pattern. And as we can see, the right side is a two-term expression <i>of perfect squares!</i> Expressions like (a+b) and (a-b) are called conjugates. So to rationalize a two-term denominator all we have to do is multiply the numerator and denominator by the conjugate of the denominator.<br>
Our denominator is {{{2sqrt(2) + 6}}}. Its conjugate is {{{2sqrt(2) - 6}}}. So this is what we will use to multiply the numerator and denominator:
{{{(7/(2sqrt(2) + 6))((2sqrt(2) - 6)/(2sqrt(2) - 6))}}}
Multiplying we get:
{{{(14sqrt(2) - 42)/((2sqrt(2))^2 - (6)^2)}}}
which simplifies to:
{{{(14sqrt(2) - 42)/(4*2 - 36)}}}
{{{(14sqrt(2) - 42)/(8 - 36)}}}
{{{(14sqrt(2) - 42)/(-24)}}}
And last of all, we reduce the fraction. Factor out 2 in the numerator:
{{{(2(7sqrt(2) - 21))/(2(-12))}}}
Cancel the 2's:
{{{(7sqrt(2) - 21)/(-12)}}}
or
{{{-(7sqrt(2) - 21)/12}}}
or
{{{(-7sqrt(2) + 21)/12}}}