Solving for x in (x^2 2sqrt[5]{frac{5^5}{10^4}1})

When solving mathematical equations, understanding the steps involved to simplify and find the value of unknown variables is essential. Let's walk through the process of solving for x in the equation:

The Equation

The given equation is:

x^2 2sqrt[5]{frac{5^5}{10^4}1}

Simplifying the Expression inside the Fifth Root

To solve for x, we need to simplify the expression inside the fifth root first. Let's break it down step by step.

Step 1: Rewrite the Denominator

We start by rewriting the denominator of the fraction inside the fifth root:

[ frac{5^5}{10^4} frac{5^5}{5^4 cdot 2^4} frac{5^5 cdot 1}{5^4 cdot 2^4} frac{5^5}{5^4 cdot 2^4} frac{5^5}{5^4 cdot 2^4} frac{5}{2^4} frac{5}{16} ]

So, the equation becomes:

x^2 2sqrt[5]{frac{5}{16}1}

Step 2: Combine and Simplify

Next, we combine and simplify the expression inside the fifth root:

[ frac{5}{16}1 frac{5}{16} 1 frac{5}{16} frac{16}{16} frac{21}{16} ]

The equation now simplifies to:

x^2 2sqrt[5]{frac{21}{16}}

Step 3: Isolate x

To isolate x, take the square root of both sides:

x pm 2^{1/2}sqrt[5]{frac{21}{16}}

However, since typically we are looking for the positive solution, we have:

x 2^{1/2}sqrt[5]{frac{21}{16}} - 2

Final Simplification

To further simplify, let's break down the expression:

x 2^{1/2} cdot sqrt[5]{frac{21}{16}} - 2

This can be written as:

x 2sqrt[5]{frac{21}{16}} - 2

Using a calculator to find the value numerically:

x 0.11179dots

Conclusion

Therefore, the value of x in the given equation is:

x 2sqrt[5]{frac{21}{16}} - 2 0.11179dots

Key Takeaways

This step-by-step process demonstrates the importance of algebraic manipulation and simplification to solve equations. The following are key concepts to remember:

Simplification: Breaking down complex expressions into simpler components. Email Address: [youremail@] Copyright: [? 2023 ]