Which of these numbers have a larger value?

1)

$\sqrt{200 -17 \sqrt{ 200 -17 \sqrt {200 -17 \sqrt{ 200 -17 \ldots}}}}$

2)

$\sqrt{200 \sqrt{ 200 \sqrt {200 \sqrt{ 200 \ldots}}}}$

Let $a$ be the value of the first nested radical, and $b$ be the values of the second nested radical.

If $a$ is greater provide the answer as, $(a+b)^{2}$ , if $b$ is greater provide the answer as $(a-b)^{2}$ .

The answer is 36864.

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Let's first solve $a$ ,

Let, $\large \sqrt{200 -17 \sqrt{ 200 -17 \sqrt {200 -17 \sqrt{ 200 -17 \ldots}}}} = x\$

Therefore, $\large \sqrt{200 -17x} = \ x$ $200 -17x = x^{2}$ $x^{2}+17x-200=0$ $x^{2}+25x-8x-200=0$ $x(x+25)-8(x+25)=0$

Hence the two values of $x$ are $x=(-25)$ and $x=8$ . We can ignore the negative value, hence $\boxed{x=8}$

Now, let's take $b$

Let, $\large \sqrt{200 \sqrt{ 200 \sqrt {200 \sqrt{ 200 \ldots}}}} = \ y$

Therefore, $\large \sqrt{200y} = \ y$ $\large 200y = \ y^{2}$ $\large 200 = \ y$

Hence the value of $\boxed{y=200}$ .

Now, as $b$ is greater than $a$ the answer is $(a-b)^{2}$ $(8-200)^{2}$ $(-192)^{2}=\boxed{\boxed{36864}}$