Video Transcript
Given that 𝑥 equals 52 over two
root 19 minus two root six and 𝑦 is the conjugate of 𝑥, find the value of 𝑥
squared 𝑦 squared.
Recall that the conjugate of an
expression of the form 𝑎 root 𝑏 plus 𝑐 root 𝑑 is 𝑎 root 𝑏 minus 𝑐 root
𝑑. Notice that 𝑥 is not currently in
this form.
In order to convert 𝑥 into the
form 𝑎 root 𝑏 plus 𝑐 root 𝑑, let’s rationalize the denominator. We do this by multiplying the
numerator and the denominator by the conjugate of the denominator, that is, by two
root 19 plus two root six over two root 19 plus two root six. We have 52 times two root 19 plus
two root six over two root 19 minus two root six times two root 19 plus two root
six, which we now simplify.
In the numerator, we have 104 root
19 plus 104 root six. And expanding the denominator, two
root 19 multiplies with itself to give four times 19. Two root six multiplies with itself
to give four times six. And the two cross terms cancel, as
desired.
The denominator has been
rationalized, that is to say, converted from an irrational to a rational number. We now have 104 root 19 plus 104
root six all over 52. Notice that this 52 is a common
factor in the numerator and the denominator.
After this final simplification,
let’s clear away our workings to make space. Since 𝑥 is equal to two root 19
plus two root six and 𝑦 is its conjugate, we can see that 𝑦 equals two root 19
minus two root six.
We can now proceed with the
calculation. Take a moment to notice though that
if we first square 𝑥 and then square 𝑦 and then multiply the results, we’re going
to end up with lots of horrible square roots all over the place. A much better plan is to first use
the law of indices, which tells us that 𝑥 squared 𝑦 squared is the same as 𝑥𝑦
squared. Because 𝑦 is conjugate to 𝑥, we
know that 𝑥𝑦 will be a rational number. In fact, we’ve already done this
calculation; it’s 52.
Our final answer is therefore 52
squared, which is 2704.
