To solve the given expression, we need to evaluate each part separately and then combine them according to the prescribed operations:
[tex]\[
\sqrt[3]{\frac{4096}{64}} + 2 \sqrt[3]{\frac{5932}{216}} - 3 \sqrt[3]{\frac{3375}{125}} + 4 \sqrt[3]{\frac{1728}{64}}
\][/tex]
Let's break it down step-by-step:
### Step 1: Evaluate [tex]\(\sqrt[3]{\frac{4096}{64}}\)[/tex]
1. Simplify the fraction:
[tex]\[
\frac{4096}{64} = 64
\][/tex]
2. Find the cube root:
[tex]\[
\sqrt[3]{64} = 4
\][/tex]
### Step 2: Evaluate [tex]\(2 \sqrt[3]{\frac{5932}{216}}\)[/tex]
1. Simplify the fraction:
[tex]\[
\frac{5932}{216} \approx 27.48148148
\][/tex]
2. Find the cube root:
[tex]\[
\sqrt[3]{27.48148148} \approx 3.0170496954707445
\][/tex]
3. Multiply by 2:
[tex]\[
2 \times 3.0170496954707445 = 6.034099390941489
\][/tex]
### Step 3: Evaluate [tex]\(3 \sqrt[3]{\frac{3375}{125}}\)[/tex]
1. Simplify the fraction:
[tex]\[
\frac{3375}{125} = 27
\][/tex]
2. Find the cube root:
[tex]\[
\sqrt[3]{27} = 3
\][/tex]
3. Multiply by 3:
[tex]\[
3 \times 3 = 9
\][/tex]
### Step 4: Evaluate [tex]\(4 \sqrt[3]{\frac{1728}{64}}\)[/tex]
1. Simplify the fraction:
[tex]\[
\frac{1728}{64} = 27
\][/tex]
2. Find the cube root:
[tex]\[
\sqrt[3]{27} = 3
\][/tex]
3. Multiply by 4:
[tex]\[
4 \times 3 = 12
\][/tex]
### Step 5: Combine the results according to the expression
Now add and subtract the results as indicated in the expression:
[tex]\[
4 + 6.034099390941489 - 9 + 12
\][/tex]
Simplify the sum:
[tex]\[
4 + 6.034099390941489 = 10.034099390941489
\][/tex]
[tex]\[
10.034099390941489 - 9 = 1.034099390941489
\][/tex]
[tex]\[
1.034099390941489 + 12 = 13.034099390941488
\][/tex]
Thus, the final result is:
[tex]\[
13.034099390941488
\][/tex]
So, the closest option from the given choices is:
(2) 13
