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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 $$a+b+c$$?

Jul 19, 2020

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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

$$4 + 14 + 21 = 39$$

Jul 19, 2020