# How do I simplify this fraction with radicals?

• MHB
• anemone
In summary, 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, leading to new discoveries and advancements. There are various ways to participate in a Simplification Challenge, such as joining a team or competition, creating your own challenge, or using simplification techniques in your own research. To be successful in a Simplification Challenge, one needs critical thinking skills, problem-solving abilities, creativity, and a good understanding of the subject matter. Simplification can be applied to all areas of science, as it allows for a better understanding and can lead to new
anemone
Gold Member
MHB
POTW Director
Simplify $\large \dfrac{2}{\sqrt{4-3\sqrt[4]{5}+2\sqrt{5}-\sqrt[4]{125}}}$.

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:

$$\displaystyle \frac{2}{\sqrt{4-3\sqrt[4]{5}+2\sqrt{5}-\sqrt[4]{125}}}=1+\sqrt[4]{5}$$

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:

$$\displaystyle \frac{2}{\sqrt{4-3\sqrt[4]{5}+2\sqrt{5}-\sqrt[4]{125}}}=1+\sqrt[4]{5}$$

Thanks for participating, MarkFL! You're very smart in checking if your first trial over the equality holds and if it does, what value of $a$ does it takes. Well done!

## 1. What is a "Simplification Challenge"?

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.

## 2. Why is simplification important in science?

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.

## 3. How can I participate in a Simplification Challenge?

There are various ways to participate in a Simplification Challenge. You can join a team or competition organized by a scientific institution or group, or you can create your own challenge with colleagues or classmates. You can also take part in online challenges or use simplification techniques in your own research.

## 4. What skills are needed for a successful Simplification Challenge?

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.

## 5. Can simplification be applied to all areas of science?

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.

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