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anemone
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Simplify $\large \dfrac{2}{\sqrt{4-3\sqrt[4]{5}+2\sqrt{5}-\sqrt[4]{125}}}$.
MarkFL said:My solution:
Let \(\displaystyle x=\sqrt[4]{5}\)
I would try to see if there is a solution for $a$ in:
\(\displaystyle \left(1+ax \right)^2\left(4-3x+2x^2-x^3 \right)=4\)
\(\displaystyle -a^2x^5+(2a^2-2a)x^4+(-3a^2+4a-1)x^3+(4a^2-6a+2)x^2+(8a-3)x+4=4\)
Now, we have:
\(\displaystyle x^5=5x\) and \(\displaystyle x^4=5\) hence we may write (after factoring):
\(\displaystyle (a-1)(1-3a)x^3+2(a-1)(2a-1)x^2+(a-1)(3-5a)x+10a(a-1)=0\)
Thus, we find $a=1$. And so we may write:
\(\displaystyle \frac{2}{\sqrt{4-3\sqrt[4]{5}+2\sqrt{5}-\sqrt[4]{125}}}=1+\sqrt[4]{5}\)
A Simplification Challenge is an exercise or activity where individuals or teams are given a complex problem or task, and are challenged to find a simpler and more efficient solution.
Simplification is important in science because it allows for a clearer understanding of complex systems and processes. By breaking down complex ideas and concepts into simpler components, scientists can more easily identify patterns and make connections, leading to new discoveries and advancements.
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To be successful in a Simplification Challenge, one needs critical thinking skills, problem-solving abilities, and creativity. It also helps to have a good understanding of the subject matter and the ability to communicate complex ideas in a simplified manner.
Yes, simplification can be applied to all areas of science, from biology and chemistry to physics and engineering. Any complex system or process can benefit from simplification, as it allows for a better understanding and can lead to new insights and innovations.