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Find the fourth root:

[tex]\[ \frac{2 \sqrt{3+2} i}{\sqrt{3}-2} \][/tex]


Sagot :

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].