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

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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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