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# Help

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Simplify: $\frac{1}{\sqrt{2}+\frac{1}{\sqrt{8}+\sqrt{200}+\frac{1}{\sqrt{18}}}}$.

Jun 13, 2019

#1
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$$\frac{1}{\sqrt{2}+\frac{1}{\sqrt{8}+\sqrt{200}+\frac{1}{\sqrt{18}}}}$$

First, simplify the radicles in the expression

$$\sqrt{8}=\sqrt{2 \cdot 2\cdot 2} =2\sqrt{2}$$

$$\sqrt{200}=\sqrt{10\cdot20\cdot2}=10\sqrt{2}$$

$$\sqrt{18}=\sqrt{3\cdot3\cdot2}=3\sqrt{2}$$

$$\frac{1}{\sqrt{2}+\frac{1}{2\sqrt{2}+10\sqrt{2}+\frac{1}{3\sqrt{2}}}}$$

The two radicles can be added together because they have the same base

$$\frac{1}{\sqrt{2}+\frac{1}{12\sqrt{2}+\frac{1}{3\sqrt{2}}}}$$

Now start simplifying from the innermost fraction

Find a common denominator to add the two on the bottom

$$\frac{1}{\sqrt{2}+\frac{1}{\frac{73}{3\sqrt{2}}}}$$

$$\frac{1}{\sqrt{2}+\frac{3\sqrt{2}}{73}}$$

Find a common denominator again

$$\frac{1}{\frac{73\sqrt{2}}{73}+\frac{3\sqrt{2}}{73}}$$

$$\frac{1}{\frac{76\sqrt{2}}{73}}$$

$$\frac{73}{76\sqrt{2}}$$

Multiply by $$\frac{\sqrt{2}}{\sqrt{2}}$$ to get the radicle out of the denominator

$$\boxed{\frac{73\sqrt{2}}{152}}$$

Jun 14, 2019
edited by power27  Jun 14, 2019
#2
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You got the answer right at this step:

73 / 76 * sqrt(2) . How did you "factor out" 73?

Jun 14, 2019
#3
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Oh oops thanks for catching that you can't do that.

power27  Jun 14, 2019