Evaluate √√√√√√√0.2

Steps to Solve

1. Identify the nested radical expression

The first step is to write down the expression for the nested radical. Based on your problem, the expression might look something like this:

$$ x = \sqrt{0.2 + \sqrt{0.2 + \sqrt{0.2 + \cdots}}} $$

This indicates that $x$ is defined in terms of itself.

2. Set up the equation

Since $x$ equals the nested radical, we can express it as:

$$ x = \sqrt{0.2 + x} $$

This equation states that the nested radical is equal to $x$.

3. Square both sides to eliminate the square root

Now, to simplify the equation, we will square both sides:

$$ x^2 = 0.2 + x $$

4. Rearrange the equation

Next, rearranging the equation gives:

$$ x^2 - x - 0.2 = 0 $$

5. Use the quadratic formula to solve for x

We can solve the quadratic equation using the quadratic formula:

$$ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$

In our equation, $a = 1$, $b = -1$, and $c = -0.2$.

6. Calculate the discriminant

First, calculate the discriminant:

$$ b^2 - 4ac = (-1)^2 - 4(1)(-0.2) = 1 + 0.8 = 1.8 $$

7. Substitute into the quadratic formula

Now substitute the values into the quadratic formula:

$$ x = \frac{-(-1) \pm \sqrt{1.8}}{2(1)} $$

Which simplifies to:

$$ x = \frac{1 \pm \sqrt{1.8}}{2} $$

8. Calculate the final values

Lastly, find the two possible values of $x$:

$$ x_1 = \frac{1 + \sqrt{1.8}}{2}, \quad x_2 = \frac{1 - \sqrt{1.8}}{2} $$

Since $x$ represents a length (being under a square root), we discard the negative solution:

$$ x = \frac{1 + \sqrt{1.8}}{2} $$

9. Evaluate the expression

Using a calculator, determine the value of $x$:

$$ \sqrt{1.8} \approx 1.34 $$

Then,

$$ x \approx \frac{1 + 1.34}{2} \approx 1.17 $$