Video Transcript
Given that nine times the square
root of eight plus 10 times the square root of 50 is equal to 𝑥 times the square
root of two, find the value of 𝑥.
In this question, we are given an
equation involving an unknown value of 𝑥 and asked to solve for 𝑥. There are a few different ways of
doing this, and we will only go through one of these. We first note that we can find 𝑥
by rewriting the left-hand side of the equation to be a multiple of the square root
of two. To do this, we can recall that if
𝑎 and 𝑏 are nonnegative real numbers, then the square root of 𝑎 squared times 𝑏
is equal to 𝑎 times the square root of 𝑏. This result will allow us to reduce
the size of the radicands by finding nontrivial square factors of the radicand.
We first note that eight is equal
to two squared times two, allowing us to rewrite the first term on the left-hand
side of the equation. Similarly, we can calculate that 50
is equal to five squared times two, allowing us to rewrite the second term on the
left-hand side of the equation as shown. The sum of these two terms must be
equal to 𝑥 times the square root of two.
We can now apply our result to
simplify the left-hand side of the equation. First, we set 𝑎 equal to two and
𝑏 equal to two to see that the square root of eight is equal to two times the
square root of two. Remember, we need to multiply this
expression by the coefficient, which is nine. Similarly, we can also set 𝑎 equal
to five and 𝑏 equal to two to see that the square root of 50 is equal to five times
the square root of two. We need to multiply this term by
10. And then we note that the sum of
these two terms is equal to 𝑥 times the square root of two.
We can now evaluate the products in
each term in any order we want, since the multiplication of real numbers is
associative. We can calculate that nine times
two is 18 and 10 times five is equal to 50. So we have that 18 root two plus 50
root two is equal to 𝑥 root two. We can then simplify the left-hand
side of the equation by noting that both terms share a factor of root two and
applying the distributive property of the multiplication of real numbers over
addition. This gives us that 18 plus 50
multiplied by the square root of two is equal to 𝑥 times the square root of
two.
We can then evaluate the sum in the
coefficient to obtain 68 root two is equal to 𝑥 root two. Finally, we can divide both sides
of the equation by the square root of two, which is the same as multiplying by the
inverse of root two, to obtain that 𝑥 is equal to 68.
