Answer :
Certainly! Let's break down the problem step-by-step to find the sum of the given expressions.
### Step 1: Simplifying each term
First, we need to simplify each term individually.
#### Term 1: [tex]\(\sqrt[3]{125 x^{10} y^{13}}\)[/tex]
- The cube root of [tex]\(125\)[/tex] is [tex]\(5\)[/tex] since [tex]\(5^3 = 125\)[/tex].
- For the variables inside the cube root:
[tex]\[ \sqrt[3]{x^{10}} = x^{\frac{10}{3}} \quad \text{and} \quad \sqrt[3]{y^{13}} = y^{\frac{13}{3}} \][/tex]
Combining these,
[tex]\[ \sqrt[3]{125 x^{10} y^{13}} = 5 x^{\frac{10}{3}} y^{\frac{13}{3}} \][/tex]
#### Term 2: [tex]\(\sqrt[3]{27 x^{10} y^{13}}\)[/tex]
- The cube root of [tex]\(27\)[/tex] is [tex]\(3\)[/tex] since [tex]\(3^3 = 27\)[/tex].
- For the variables,
[tex]\[ \sqrt[3]{x^{10}} = x^{\frac{10}{3}} \quad \text{and} \quad \sqrt[3]{y^{13}} = y^{\frac{13}{3}} \][/tex]
Combining these,
[tex]\[ \sqrt[3]{27 x^{10} y^{13}} = 3 x^{\frac{10}{3}} y^{\frac{13}{3}} \][/tex]
#### Term 3: [tex]\(8 x^3 y^4 (\sqrt[3]{x y})\)[/tex]
- [tex]\(\sqrt[3]{x y}\)[/tex] can be written as [tex]\(x^{\frac{1}{3}} y^{\frac{1}{3}}\)[/tex].
Thus,
[tex]\[ 8 x^3 y^4 (\sqrt[3]{x y}) = 8 x^3 y^4 \cdot x^{\frac{1}{3}} y^{\frac{1}{3}} = 8 x^{3 + \frac{1}{3}} y^{4 + \frac{1}{3}} = 8 x^{\frac{10}{3}} y^{\frac{13}{3}} \][/tex]
#### Term 4: [tex]\(15 x^6 y^8 (\sqrt[3]{x y})\)[/tex]
Similarly,
[tex]\[ 15 x^6 y^8 (\sqrt[3]{x y}) = 15 x^6 y^8 \cdot x^{\frac{1}{3}} y^{\frac{1}{3}} = 15 x^{6 + \frac{1}{3}} y^{8 + \frac{1}{3}} = 15 x^{\frac{19}{3}} y^{\frac{25}{3}} \][/tex]
#### Term 5: [tex]\(15 x^3 y^4 (\sqrt[3]{x y})\)[/tex] again simplifies to:
[tex]\[ 15 x^{\frac{10}{3}} y^{\frac{13}{3}} \][/tex]
#### Term 6: [tex]\(8 x^6 y^8 (\sqrt[3]{x y})\)[/tex]
Same simplification as Term 4:
[tex]\[ 8 x^6 y^8 (\sqrt[3]{x y}) = 8 x^{\frac{19}{3}} y^{\frac{25}{3}} \][/tex]
### Step 2: Summing all the simplified terms
Now let's collect and add up these simplified terms:
[tex]\[ \sqrt[3]{125 x^{10} y^{13}} + \sqrt[3]{27 x^{10} y^{13}} + 8 x^3 y^4 (\sqrt[3]{x y}) + 15 x^6 y^8 (\sqrt[3]{x y}) + 15 x^3 y^4 (\sqrt[3]{x y}) + 8 x^6 y^8 (\sqrt[3]{x y}) \][/tex]
This simplifies into types [tex]\(x^{\frac{10}{3}} y^{\frac{13}{3}} \)[/tex] and [tex]\( x^{\frac{19}{3}} y^{\frac{25}{3}}\)[/tex]:
Combining these terms:
[tex]\[ 5 x^{\frac{10}{3}} y^{\frac{13}{3}} + 3 x^{\frac{10}{3}} y^{\frac{13}{3}} + 8 x^{\frac{10}{3}} y^{\frac{13}{3}} + 15 x^{\frac{19}{3}} y^{\frac{25}{3}} + 15 x^{\frac{10}{3}} y^{\frac{13}{3}} + 8 x^{\frac{19}{3}} y^{\frac{25}{3}} \][/tex]
### Step 3: Grouping and Final Sum
Grouping the terms with similar exponents:
[tex]\[ (5 + 3 + 8 + 15) x^{\frac{10}{3}} y^{\frac{13}{3}} + (8 + 15) x^{\frac{19}{3}} y^{\frac{25}{3}} \][/tex]
Simplifies to:
[tex]\[ 31 x^{\frac{10}{3}} y^{\frac{13}{3}} + 23 x^{\frac{19}{3}} y^{\frac{25}{3}} \][/tex]
So the overall sum is:
[tex]\[ 31 x^{\frac{10}{3}} y^{\frac{13}{3}} + 23 x^{\frac{19}{3}} y^{\frac{25}{3}} \][/tex]
This is the detailed solution to the given problem.
### Step 1: Simplifying each term
First, we need to simplify each term individually.
#### Term 1: [tex]\(\sqrt[3]{125 x^{10} y^{13}}\)[/tex]
- The cube root of [tex]\(125\)[/tex] is [tex]\(5\)[/tex] since [tex]\(5^3 = 125\)[/tex].
- For the variables inside the cube root:
[tex]\[ \sqrt[3]{x^{10}} = x^{\frac{10}{3}} \quad \text{and} \quad \sqrt[3]{y^{13}} = y^{\frac{13}{3}} \][/tex]
Combining these,
[tex]\[ \sqrt[3]{125 x^{10} y^{13}} = 5 x^{\frac{10}{3}} y^{\frac{13}{3}} \][/tex]
#### Term 2: [tex]\(\sqrt[3]{27 x^{10} y^{13}}\)[/tex]
- The cube root of [tex]\(27\)[/tex] is [tex]\(3\)[/tex] since [tex]\(3^3 = 27\)[/tex].
- For the variables,
[tex]\[ \sqrt[3]{x^{10}} = x^{\frac{10}{3}} \quad \text{and} \quad \sqrt[3]{y^{13}} = y^{\frac{13}{3}} \][/tex]
Combining these,
[tex]\[ \sqrt[3]{27 x^{10} y^{13}} = 3 x^{\frac{10}{3}} y^{\frac{13}{3}} \][/tex]
#### Term 3: [tex]\(8 x^3 y^4 (\sqrt[3]{x y})\)[/tex]
- [tex]\(\sqrt[3]{x y}\)[/tex] can be written as [tex]\(x^{\frac{1}{3}} y^{\frac{1}{3}}\)[/tex].
Thus,
[tex]\[ 8 x^3 y^4 (\sqrt[3]{x y}) = 8 x^3 y^4 \cdot x^{\frac{1}{3}} y^{\frac{1}{3}} = 8 x^{3 + \frac{1}{3}} y^{4 + \frac{1}{3}} = 8 x^{\frac{10}{3}} y^{\frac{13}{3}} \][/tex]
#### Term 4: [tex]\(15 x^6 y^8 (\sqrt[3]{x y})\)[/tex]
Similarly,
[tex]\[ 15 x^6 y^8 (\sqrt[3]{x y}) = 15 x^6 y^8 \cdot x^{\frac{1}{3}} y^{\frac{1}{3}} = 15 x^{6 + \frac{1}{3}} y^{8 + \frac{1}{3}} = 15 x^{\frac{19}{3}} y^{\frac{25}{3}} \][/tex]
#### Term 5: [tex]\(15 x^3 y^4 (\sqrt[3]{x y})\)[/tex] again simplifies to:
[tex]\[ 15 x^{\frac{10}{3}} y^{\frac{13}{3}} \][/tex]
#### Term 6: [tex]\(8 x^6 y^8 (\sqrt[3]{x y})\)[/tex]
Same simplification as Term 4:
[tex]\[ 8 x^6 y^8 (\sqrt[3]{x y}) = 8 x^{\frac{19}{3}} y^{\frac{25}{3}} \][/tex]
### Step 2: Summing all the simplified terms
Now let's collect and add up these simplified terms:
[tex]\[ \sqrt[3]{125 x^{10} y^{13}} + \sqrt[3]{27 x^{10} y^{13}} + 8 x^3 y^4 (\sqrt[3]{x y}) + 15 x^6 y^8 (\sqrt[3]{x y}) + 15 x^3 y^4 (\sqrt[3]{x y}) + 8 x^6 y^8 (\sqrt[3]{x y}) \][/tex]
This simplifies into types [tex]\(x^{\frac{10}{3}} y^{\frac{13}{3}} \)[/tex] and [tex]\( x^{\frac{19}{3}} y^{\frac{25}{3}}\)[/tex]:
Combining these terms:
[tex]\[ 5 x^{\frac{10}{3}} y^{\frac{13}{3}} + 3 x^{\frac{10}{3}} y^{\frac{13}{3}} + 8 x^{\frac{10}{3}} y^{\frac{13}{3}} + 15 x^{\frac{19}{3}} y^{\frac{25}{3}} + 15 x^{\frac{10}{3}} y^{\frac{13}{3}} + 8 x^{\frac{19}{3}} y^{\frac{25}{3}} \][/tex]
### Step 3: Grouping and Final Sum
Grouping the terms with similar exponents:
[tex]\[ (5 + 3 + 8 + 15) x^{\frac{10}{3}} y^{\frac{13}{3}} + (8 + 15) x^{\frac{19}{3}} y^{\frac{25}{3}} \][/tex]
Simplifies to:
[tex]\[ 31 x^{\frac{10}{3}} y^{\frac{13}{3}} + 23 x^{\frac{19}{3}} y^{\frac{25}{3}} \][/tex]
So the overall sum is:
[tex]\[ 31 x^{\frac{10}{3}} y^{\frac{13}{3}} + 23 x^{\frac{19}{3}} y^{\frac{25}{3}} \][/tex]
This is the detailed solution to the given problem.
