Video Transcript
Given that 𝑥 is equal to the
square root of 13 minus three and 𝑦 is equal to the square root of 13 plus three,
find the value of 𝑥 times 𝑦 raised to the power of negative one plus 𝑦 times 𝑥
raised to the power of negative one.
In this question, we are given
radical expressions for 𝑥 and 𝑦 and asked to evaluate an expression involving 𝑥
and 𝑦. To evaluate this expression, we can
start by recalling that raising a number to the power of negative one is the same as
taking its reciprocal. So our expression is equal to 𝑥
over 𝑦 plus 𝑦 over 𝑥.
We can then substitute the values
of 𝑥 and 𝑦 into the expression to obtain root 13 minus three over root 13 plus
three plus root 13 plus three over root 13 minus three. To add these terms together, we
want them to have the same denominator. We can do this by rationalizing
both denominators.
Let’s start by rationalizing the
denominator in the first term. We can multiply the numerator and
denominator of this term by the conjugate of the denominator. That is, we multiply by root 13
minus three over root 13 minus three. We can then note that in the
denominator, we have a product in the form of a difference of two squares, where 𝑎
is equal to root 13 and 𝑏 is equal to three. Therefore, the denominator of this
term is root 13 squared minus three squared, which is equal to 13 minus nine.
We can then clear some space and
note that we have written the first term to be equal to root 13 minus three all
squared all divided by four. We can then apply the same process
to the second term. We want to multiply the numerator
and denominator by a conjugate of the denominator. In this case, we will multiply by
root 13 plus three over root 13 plus three. We then evaluate to obtain root 13
plus three all squared all over four. Hence, we have rewritten the
expression as root 13 minus three all squared all over four plus root 13 plus three
all squared all over four.
We now want to add these
expressions together. We note that the denominators are
equal, so we can add the numerators. We will do this by using the fact
that 𝑎 plus 𝑏 all squared is equal to 𝑎 squared plus two 𝑎𝑏 plus 𝑏
squared. And 𝑎 minus 𝑏 all squared is
equal to 𝑎 squared minus two 𝑎𝑏 plus 𝑏 squared.
We can start by expanding the
product in the numerator of the first term. We get root 13 squared minus two
times root 13 times three plus three squared all over four. We can then evaluate each term in
the numerator to obtain 13 minus six root 13 plus nine all over four. We can then add the integers in the
numerator to get 22 minus six root 13 all over four. We could cancel a shared factor of
two. However, we will leave the
expression as it is to keep the denominators of the terms the same.
We now want to apply the same
process to the second term. We expand the numerator using the
binomial formula to obtain root 13 squared plus two times root 13 times three plus
three squared all over four. We can then evaluate the exponents
and simplify to get 22 plus six root 13 all over four. Once again, we will leave the
denominator as four. Therefore, our expression is equal
to 22 minus six root 13 over four plus 22 plus six root 13 over four.
Since the denominators are the
same, we can now just add the numerators. We can then note that negative six
root 13 plus six root 13 is equal to zero. So we are left with 44 over four,
which we can then evaluate is equal to 11. Hence, if 𝑥 is root 13 minus three
and 𝑦 is root 13 plus three, then 𝑥 times 𝑦 to the power of negative one plus 𝑦
times 𝑥 to the power of negative one is equal to 11.
