Algebra

Radical Expressions and Equations

Radical Expressions and Equations: Level 4 Challenges

         

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

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

If x+34x1+x+86x1=1\sqrt{x+3-4\sqrt{x-1}}+\sqrt{x+8-6\sqrt{x-1}}=1 what is the sum of all integer values that xx can take?

xx2+78x8+1364=179x-\sqrt{\frac{x}{2}+\frac{7}{8}-\sqrt{\frac{x}{8}+\frac{13}{64}}}=179

Find the value of 10x10x.

Inspiration

113+23+43+143+63+93+193+123+163 \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 a3+b{\sqrt[3] a + b} for integers aa and bb, find the value of a+ba+b.

Source: MAΘMA\Theta 1992.

The minimum value of x4x224x+145+x423x22x+145\sqrt{x^4 - x^2 - 24x + 145} + \sqrt{x^4 - 23x^2 - 2x + 145} can be expressed in the form aba\sqrt{b} , where aa and bb are integers, with bb is not divisible by the square of any prime. What is the value of a+ba+b ?

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