Video Transcript
Without using a calculator, find
two consecutive integers that root 13 plus root six lies between.
Since 13 lies between the perfect
squares nine and 16, we can take square roots and find that root 13 lies between
three and four. Similarly, six lies between four
and nine, and so root six lies between two and three. We can combine these inequalities
to force root 13 plus root six between five and seven. The question therefore hinges on
whether root 13 plus root six is greater than or smaller than six. We can work this out by squaring
root 13 plus root six and comparing this quantity to six squared.
Expanding the parentheses, we have
root 13 plus root six all squared equals root 13 squared plus two root 13 times root
six plus root six squared, which equals 19 plus two root 78. It is this quantity that we want to
compare to six squared, or 36. 19 plus two root 78 is either
greater than or less than 36. We don’t yet know which. Let’s represent this relationship
for the moment by an orange question mark. Whether the quantity on the left is
greater than or less than the quantity on the right will not be changed by
subtracting 19 from both nor by dividing both quantities by two. We have reduced to the question of
whether root 78 is greater than or less than 8.5.
Since both of these numbers are in
any case positive, we can answer this question by squaring them. It is a little irritating having to
square 8.5 without a calculator, but we can do it. It’s eight times eight plus a half
times eight plus another half times eight plus a half times a half. And this is 72.25, which is less
than 78. This tells us that root 13 plus
root six all squared, which equals 19 plus two root 78, is greater than 36, which is
six squared. It follows that root 13 plus root
six is greater than six.
We have found that root 13 plus
root six lies between the consecutive integers six and seven.
