Video Transcript
Given that 𝑥 is equal to root
seven divided by root 13 and 𝑦 is equal to root 13 divided by root seven, find 182
multiplied by 𝑥 plus 𝑦, expressing your answer in simplest form.
Our first step here is to
rationalize the denominator of both 𝑥 and 𝑦. In order to rationalize the
denominator of 𝑥, we multiply the top and bottom of the fraction by root 13. At this point, it is worth
remembering two of the laws of surds. Root 𝑎 multiplied by root 𝑎 is
equal to 𝑎, and root 𝑎 multiplied by root 𝑏 is equal to root 𝑎𝑏.
In our question, root seven
multiplied by root 13 will be equal to root 91, as seven multiplied by 13 equals
91. Multiplying the denominators gives
us 13, as root 13 multiplied by root 13 is equal to 13.
𝑦 was equal to root 13 divided by
root seven. To rationalize this surd, we
multiply the top and bottom by root seven. Once again, root 13 multiplied by
root seven is equal to root 91, and root seven multiplied by root seven is equal to
seven.
We now have simplified expressions
for 𝑥 and 𝑦. We want to calculate 182 multiplied
by 𝑥 plus 𝑦. Let’s first work out what 𝑥 plus
𝑦 would be. 𝑥 plus 𝑦 is equal to root 91
divided by 13 plus root 91 divided by seven. In order to add two fractions, we
need to find a common denominator, in this case a number that is in the seven and 13
times table. The lowest common denominator in
this case is 91. 13 multiplied by seven is equal to
91.
Multiplying the numerator and
denominator of the first fraction by seven gives a seven root 91 divided by 91. Multiplying the numerator and
denominator of the second fraction by 13 gives us 13 root 91 divided by 91.
Now that the denominators are the
same, we can add the two numerators. Seven root 91 plus 13 root 91 is
equal to 20 root 91. Therefore, 𝑥 plus 𝑦 is equal to
20 root 91 divided by 91. We now need to multiply this
expression by 182 to get the final answer in this question. 182 divided by 91 is equal to two,
as 91 is a half of 182. We’re therefore left with two
multiplied by 20 root 91. This is equal to 40 root 91. If 𝑥 is equal to root seven
divided by root 13 and 𝑦 is equal to root 13 divided by root seven, then 182
multiplied by 𝑥 plus 𝑦 is equal to 40 root 91.
