Video Transcript
Fill in the blank. The square root of five added to 10
plus the square root of seven is equal to the square root of five plus what added to
10.
In this question, we are asked to
fill in the blank in an equation. If we look at the given equation,
we can see that it involves adding real numbers. In particular, root five and 10
appear on both sides of the equation. So this seems to involve switching
the order of the sum of real numbers and switching the order of the evaluation.
Let’s start with the left-hand side
of the equation and try to write it in the same form as the right-hand side of the
equation. To do this, we need the addition by
10 to be at the end of the expression. We can first recall that the
addition of real numbers is commutative. So, for real numbers 𝑎 and 𝑏, we
have that 𝑎 plus 𝑏 is equal to 𝑏 plus 𝑎. This allows us to switch the order
of the two terms inside the parentheses. We obtain the square root of five
added to the square root of seven plus 10.
However, this is not quite in the
same form as the right-hand side of the given equation. We need the addition to root five
to be the first operation evaluated. We can then recall that we can
evaluate the sum of real numbers in any order by using the associative property of
the addition of real numbers. In general, we have that for any
real numbers 𝑎, 𝑏, and 𝑐, we have that 𝑎 added to 𝑏 plus 𝑐 is equal to 𝑎 plus
𝑏 added to 𝑐. This means that we can instead add
the square root of five to the square root of seven and then add 10. We see that this is the same form
as the right-hand side of the given equation. So we can fill in the blank with
the square root of seven.
