Algebra

# Radical Expressions and Equations: Level 4 Challenges

$\sqrt { ab+56 } -\sqrt { ab-56 } =4$

If all the $n$ pairs of positive integers $(a,b)$ that satisfy the equation above are

$(a_1 ,b_1) , (a_2, b_2) , \ldots , (a_n, b_n) ,$

submit your answer as $\displaystyle \sum _{ i=1 }^{ n }{ a_{ i }b_{ i } }$.

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 $a$ and $b$ are integers, with $b$ is not divisible by the square of any prime. What is the value of $a+b$ ?

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