+367 The ratio of the areas of two squares is \(\frac{32}{63}\). After rationalizing the denominator, the ratio of their side lengths can be expressed in the simplified form \(\frac{a\sqrt{b}}{c}\) where a, b, and c are integers. What is the value of the sum \(a+b+c \)?
+1094 First, we need to find the side lengths. Since the side length of a square is the square root of the area, we now know that the two side lengths are
\(\sqrt {32} \text{ and } \sqrt{63}\)
The ratio would now be \(\frac{\sqrt{32}}{\sqrt{63}}\).
Simplifying the square roots, we get \(\frac{4\sqrt 2}{3\sqrt 7}\).
From here, we need to multiply the top and bottom by \(\sqrt 7\) to rationalize the denominator. So, we get
\(\frac{4\sqrt {14}}{21}\)
Now, all you have to do is add a, b, and c. This will give us
\(4 + 14 + 21 = 39\)
