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].
