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Combine your skills of working with exponents and manipulating expressions for radical glory.

# Level 4

Determine the sum of all possible integer values of $$x<100$$ such that $$y$$ is an integer.

$$\sqrt{x+\sqrt{x+\sqrt{x+\sqrt{x...}}}}=y$$

If $\sqrt{x+3-4\sqrt{x-1}}+\sqrt{x+8-6\sqrt{x-1}}=1$ what is the sum of all integer values that $$x$$ can take?

$x-\sqrt{\frac{x}{2}+\frac{7}{8}-\sqrt{\frac{x}{8}+\frac{13}{64}}}=179$

Find the value of $$10x$$.

Inspiration

$\Large{\frac1{\sqrt[3]{1}+\sqrt[3]{2}+\sqrt[3]{4}} + \frac1{\sqrt[3]{4}+\sqrt[3]{6}+\sqrt[3]{9}} + \frac1{\sqrt[3]{9}+\sqrt[3]{12}+\sqrt[3]{16}}}$

If the expression above can be simplified to the form of $${\sqrt[3] a + b}$$ for integers $$a$$ and $$b$$, find the value of $$a+b$$.

###### Source: $$MA\Theta$$ 1992.

The minimum value of $\sqrt{x^4 - x^2 - 24x + 145} + \sqrt{x^4 - 23x^2 - 2x + 145}$ can be expressed in the form $$a\sqrt{b}$$ , where $$a$$ is an integer, $$b$$ and is not divisible by the square of any prime. What is the value of $$a+b$$ ?

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