# Challenging Algebra Questions

Challenging algebra questions are presented along with their detailed solutions. These questions needs the basic algebra rules, exponents, complex numbers, factoring, ... , thinking and some patience. Do not give up quickly; you will be learning a lot while you are looking for the solution.

## Questions with Solutions

 

1. If $x^2 + \dfrac{1}{x^2} = 10$, then what is the exact value of $x^5 + \dfrac{1}{x^5}$ for $x \ge 0$?
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2. Write the complex number $(1-i)^{1+i}$ in standard form $a + i b$ where $i=\sqrt{-1}$ is the imaginary unit.
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3. Solve the trigonometric equation given by $\quad 6 \cos(x + \pi/4) + 8 \sin(x + \pi/4) = 5$
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4. What are the real solutions to the equation $(\sqrt {x})^{|x|} = x^{ x^2+\frac{1}{18}}$?
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5. Given that $x + y = 4$ and $x^3 + y^3 = 24$, evaluate $x^4 + y^4$
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6. Show that $\left(\dfrac{2+\sqrt{-4}}{2}\right)^{10} + \left(\dfrac{2-\sqrt{-4}}{2}\right)^{10} = 0$
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7. Solve the equation $x + x^2 + x^3 = 2 + 4 + 8$
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8. Solve for $x$ the equation: $\quad 2x^4 - x^3 - x^2(1 - 4a) - a x + 2 a^2 = 0$
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9. Simplify $\quad \dfrac{a\sqrt{b^2} - b \sqrt {a^2}}{\sqrt{(ab)^2}}$ if $a \gt 0$ and $b \lt 0$.
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