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# Chris can you halp me?

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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 $a+b+c$?

Nov 18, 2020

#1
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First, we need to find the square roots of the 2 squares, because the squares of them will give us 32 and 63. So, the square root of 32 is $$\sqrt{32}$$, which simplifies to: $$\sqrt{16} \cdot \sqrt2 \implies 4\sqrt2$$

We can use this same thing to find the square root of 63.

$$\sqrt{63} \implies \sqrt9 \cdot \sqrt7 \implies 3\sqrt7$$

So when finding the square roots, our fraction becomes $$\frac{\sqrt{32}}{\sqrt{63}}$$, which using our steps makes the fraction $$\frac{4\sqrt2}{3\sqrt7}$$

To rationalize this, we need to multiply $$\sqrt7$$ on both sides, so this gives us:

$$\frac{4\sqrt{2} \cdot \sqrt7}{3\sqrt7 \cdot \sqrt7} \implies \frac{4\sqrt{14}}{21}$$

That makes a=4, b=14, and c=21. Adding 4+14+21 give us $$\boxed{39}$$

Nov 18, 2020
#2
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You misspelled help.

just joking

Nov 18, 2020