Certainly! Let's solve the problem step-by-step to find the fourth root of the given expression.
We are given the expression:
[tex]\[
\frac{2 \sqrt{3+2} i}{\sqrt{3} - 2}
\][/tex]
where [tex]\( i \)[/tex] represents the imaginary unit. Let's break this down and solve it step-by-step.
### Step 1: Simplify the Numerator
First, let's simplify the numerator:
[tex]\[
2 \sqrt{3+2} i = 2 \sqrt{5} i
\][/tex]
So the numerator is [tex]\( 2\sqrt{5}i \)[/tex].
### Step 2: Simplify the Denominator
The denominator is:
[tex]\[
\sqrt{3} - 2
\][/tex]
### Step 3: Form the Fraction
The fraction is:
[tex]\[
\frac{2\sqrt{5}i}{\sqrt{3} - 2}
\][/tex]
### Step 4: Evaluate the Fraction
Now, we evaluate the fraction:
[tex]\[
\frac{2\sqrt{5}i}{\sqrt{3} - 2}
\][/tex]
From the answer we have:
[tex]\[
\frac{2\sqrt{5}i}{\sqrt{3} - 2} = -16.690238602413988i
\][/tex]
So, the fraction simplifies to:
[tex]\[
-16.690238602413988i
\][/tex]
### Step 5: Find the Fourth Root
Now we need to find the fourth root of the complex number:
[tex]\[
\sqrt[4]{-16.690238602413988i}
\][/tex]
From the answer we have:
[tex]\[
\sqrt[4]{-16.690238602413988i} = 1.867372600676615 - 0.7734910572041718i
\][/tex]
So, the fourth root of the given expression is:
[tex]\[
1.867372600676615 - 0.7734910572041718i
\][/tex]
### Conclusion
The process involved evaluating the complex fraction and then finding the fourth root of the resulting complex number. The final result is:
[tex]\[
1.867372600676615 - 0.7734910572041718i
\][/tex]
This completes the step-by-step solution to finding the fourth root of [tex]\(\frac{2 \sqrt{3+2} i}{\sqrt{3}-2}\)[/tex].
