Answer :
Let's break down the problem and solve it step by step to find the total sum of the given expressions:
We need to calculate the following sum:
1. [tex]\(2\left(\sqrt[3]{16x^3y}\right)\)[/tex]
2. [tex]\(4\left(\sqrt[3]{54x^6y^5}\right)\)[/tex]
3. [tex]\(4x(\sqrt[3]{2y})\)[/tex]
4. [tex]\(12x^2y\left(\sqrt[3]{2y^2}\right)\)[/tex]
5. [tex]\(8x(\sqrt[3]{xy})\)[/tex]
6. [tex]\(12x^3y^2(\sqrt[3]{6y})\)[/tex]
7. [tex]\(16x^3y\left(\sqrt[3]{2y^2}\right)\)[/tex]
8. [tex]\(48x^3y(\sqrt[3]{2y})\)[/tex]
Let's evaluate each term one by one:
1. First term: [tex]\(2\left(\sqrt[3]{16x^3y}\right)\)[/tex]
- Calculate [tex]\(\sqrt[3]{16x^3y} = (16x^3y)^{1/3}\)[/tex].
- This term simplifies to [tex]\(2 \times (16x^3y)^{1/3}\)[/tex].
2. Second term: [tex]\(4\left(\sqrt[3]{54x^6y^5}\right)\)[/tex]
- Calculate [tex]\(\sqrt[3]{54x^6y^5} = (54x^6y^5)^{1/3}\)[/tex].
- This term becomes [tex]\(4 \times (54x^6y^5)^{1/3}\)[/tex].
3. Third term: [tex]\(4x(\sqrt[3]{2y})\)[/tex]
- Calculate [tex]\(\sqrt[3]{2y} = (2y)^{1/3}\)[/tex].
- This becomes [tex]\(4x \times (2y)^{1/3}\)[/tex].
4. Fourth term: [tex]\(12x^2y\left(\sqrt[3]{2y^2}\right)\)[/tex]
- Calculate [tex]\(\sqrt[3]{2y^2} = (2y^2)^{1/3}\)[/tex].
- This becomes [tex]\(12x^2y \times (2y^2)^{1/3}\)[/tex].
5. Fifth term: [tex]\(8x(\sqrt[3]{xy})\)[/tex]
- Calculate [tex]\(\sqrt[3]{xy} = (xy)^{1/3}\)[/tex].
- This term is [tex]\(8x \times (xy)^{1/3}\)[/tex].
6. Sixth term: [tex]\(12x^3y^2(\sqrt[3]{6y})\)[/tex]
- Calculate [tex]\(\sqrt[3]{6y} = (6y)^{1/3}\)[/tex].
- This becomes [tex]\(12x^3y^2 \times (6y)^{1/3}\)[/tex].
7. Seventh term: [tex]\(16x^3y\left(\sqrt[3]{2y^2}\right)\)[/tex]
- Calculate [tex]\(\sqrt[3]{2y^2} = (2y^2)^{1/3}\)[/tex].
- This becomes [tex]\(16x^3y \times (2y^2)^{1/3}\)[/tex].
8. Eighth term: [tex]\(48x^3y(\sqrt[3]{2y})\)[/tex]
- Calculate [tex]\(\sqrt[3]{2y} = (2y)^{1/3}\)[/tex].
- This term is [tex]\(48x^3y \times (2y)^{1/3}\)[/tex].
After evaluating and simplifying each term, we sum up all these expressions to get:
```
20.158736798318x^3y(y^2)0.333333333333333 +
60.4762103949539x^3y1.33333333333333 +
21.8054471139857x^3y2.33333333333333 +
15.1190525987385x^2y(y^2)0.333333333333333 +
5.03968419957949xy0.333333333333333 +
8x(xy)0.333333333333333 +
5.03968419957949(x^3y)0.333333333333333 +
15.1190525987385(x^6*y^5)0.333333333333333
```
These numerical approximations represent the final sum of the complex expression given the operations performed on each term.
We need to calculate the following sum:
1. [tex]\(2\left(\sqrt[3]{16x^3y}\right)\)[/tex]
2. [tex]\(4\left(\sqrt[3]{54x^6y^5}\right)\)[/tex]
3. [tex]\(4x(\sqrt[3]{2y})\)[/tex]
4. [tex]\(12x^2y\left(\sqrt[3]{2y^2}\right)\)[/tex]
5. [tex]\(8x(\sqrt[3]{xy})\)[/tex]
6. [tex]\(12x^3y^2(\sqrt[3]{6y})\)[/tex]
7. [tex]\(16x^3y\left(\sqrt[3]{2y^2}\right)\)[/tex]
8. [tex]\(48x^3y(\sqrt[3]{2y})\)[/tex]
Let's evaluate each term one by one:
1. First term: [tex]\(2\left(\sqrt[3]{16x^3y}\right)\)[/tex]
- Calculate [tex]\(\sqrt[3]{16x^3y} = (16x^3y)^{1/3}\)[/tex].
- This term simplifies to [tex]\(2 \times (16x^3y)^{1/3}\)[/tex].
2. Second term: [tex]\(4\left(\sqrt[3]{54x^6y^5}\right)\)[/tex]
- Calculate [tex]\(\sqrt[3]{54x^6y^5} = (54x^6y^5)^{1/3}\)[/tex].
- This term becomes [tex]\(4 \times (54x^6y^5)^{1/3}\)[/tex].
3. Third term: [tex]\(4x(\sqrt[3]{2y})\)[/tex]
- Calculate [tex]\(\sqrt[3]{2y} = (2y)^{1/3}\)[/tex].
- This becomes [tex]\(4x \times (2y)^{1/3}\)[/tex].
4. Fourth term: [tex]\(12x^2y\left(\sqrt[3]{2y^2}\right)\)[/tex]
- Calculate [tex]\(\sqrt[3]{2y^2} = (2y^2)^{1/3}\)[/tex].
- This becomes [tex]\(12x^2y \times (2y^2)^{1/3}\)[/tex].
5. Fifth term: [tex]\(8x(\sqrt[3]{xy})\)[/tex]
- Calculate [tex]\(\sqrt[3]{xy} = (xy)^{1/3}\)[/tex].
- This term is [tex]\(8x \times (xy)^{1/3}\)[/tex].
6. Sixth term: [tex]\(12x^3y^2(\sqrt[3]{6y})\)[/tex]
- Calculate [tex]\(\sqrt[3]{6y} = (6y)^{1/3}\)[/tex].
- This becomes [tex]\(12x^3y^2 \times (6y)^{1/3}\)[/tex].
7. Seventh term: [tex]\(16x^3y\left(\sqrt[3]{2y^2}\right)\)[/tex]
- Calculate [tex]\(\sqrt[3]{2y^2} = (2y^2)^{1/3}\)[/tex].
- This becomes [tex]\(16x^3y \times (2y^2)^{1/3}\)[/tex].
8. Eighth term: [tex]\(48x^3y(\sqrt[3]{2y})\)[/tex]
- Calculate [tex]\(\sqrt[3]{2y} = (2y)^{1/3}\)[/tex].
- This term is [tex]\(48x^3y \times (2y)^{1/3}\)[/tex].
After evaluating and simplifying each term, we sum up all these expressions to get:
```
20.158736798318x^3y(y^2)0.333333333333333 +
60.4762103949539x^3y1.33333333333333 +
21.8054471139857x^3y2.33333333333333 +
15.1190525987385x^2y(y^2)0.333333333333333 +
5.03968419957949xy0.333333333333333 +
8x(xy)0.333333333333333 +
5.03968419957949(x^3y)0.333333333333333 +
15.1190525987385(x^6*y^5)0.333333333333333
```
These numerical approximations represent the final sum of the complex expression given the operations performed on each term.
