Welcome to Westonci.ca, the place where your questions are answered by a community of knowledgeable contributors. Connect with a community of experts ready to help you find solutions to your questions quickly and accurately. Get immediate and reliable solutions to your questions from a community of experienced professionals on our platform.
Sagot :
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].
### 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].
Thanks for using our platform. We're always here to provide accurate and up-to-date answers to all your queries. We appreciate your time. Please come back anytime for the latest information and answers to your questions. Your questions are important to us at Westonci.ca. Visit again for expert answers and reliable information.