Radical Expressions and Equations

Radical Expressions and Equations: Level 4 Challenges


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


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?


Find the value of \(10x\).


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

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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