Answer :

GinnyAnswer
Certainly! Let’s break down the problem step by step.

### Step 1: Simplify the Cube Root Expression

Firstly, we are asked to simplify the term [tex]\(\sqrt[3]{-\frac{64}{27}}\)[/tex].

1. Determine the fraction inside the cube root:
[tex]\[ -\frac{64}{27} \][/tex]

2. Compute the cube root of [tex]\(-\frac{64}{27}\)[/tex]:

- The cube root of the numerator ([tex]\(-64\)[/tex]) is [tex]\(-4\)[/tex].
- The cube root of the denominator ([tex]\(27\)[/tex]) is [tex]\(3\)[/tex].

Combined, the cube root is:
[tex]\[ \sqrt[3]{-\frac{64}{27}} = \frac{\sqrt[3]{-64}}{\sqrt[3]{27}} = \frac{-4}{3} \approx -1.5874 + 1.154701j \][/tex]

### Step 2: Calculate the Square Root Expression

Next, we need to calculate [tex]\(\sqrt{81.36}\)[/tex]:
[tex]\[ \sqrt{81.36} = 9.019978 \][/tex]

### Step 3: Multiply the Results

Now, we multiply the results from Step 1 and Step 2:
[tex]\[ D = \left(0.6667 + 1.154701j\right) \times 9.019978 \][/tex]

### Step 4: Simplify the Complex Product

We multiply the complex number by the real number:
[tex]\[ D = \left(0.6667 + 1.154701j\right) \times 9.019978 \][/tex]

To find the real and imaginary parts:

- Real Part:
[tex]\[ \text{Real part} = 0.6667 \times 9.019978 = 6.013319 \][/tex]

- Imaginary Part:
[tex]\[ \text{Imaginary part} = 1.154701 \times 9.019978 = 10.415373 \][/tex]

Putting it all together:

[tex]\[ D = 6.013319 + 10.415373j \][/tex]

### Conclusion

So, the detailed computation results in:
[tex]\[ D \approx 6.013319 + 10.415373j \][/tex]

This value is the product of the cube root of [tex]\(-\frac{64}{27}\)[/tex] and the square root of [tex]\(81.36\)[/tex].

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