Answer :
To find the sum of the given expressions, we need to simplify each one individually and then add the simplified forms together. Let's go through them one by one.
### Step 1: Simplify each cubic root term
1. [tex]\(\sqrt[3]{125 x^{10} y^{13}}\)[/tex]
We can break this expression down as follows:
[tex]\[ \sqrt[3]{125 x^{10} y^{13}} = \sqrt[3]{125} \cdot \sqrt[3]{x^{10}} \cdot \sqrt[3]{y^{13}} \][/tex]
- [tex]\(\sqrt[3]{125} = 5\)[/tex] because [tex]\(5^3 = 125\)[/tex]
- [tex]\(\sqrt[3]{x^{10}} = x^{\frac{10}{3}} = x^{3 + \frac{1}{3}}\)[/tex]
- [tex]\(\sqrt[3]{y^{13}} = y^{\frac{13}{3}}\)[/tex]
So,
[tex]\[ \sqrt[3]{125 x^{10} y^{13}} = 5 x^{3 + \frac{1}{3}} y^{\frac{13}{3}} \][/tex]
2. [tex]\(\sqrt[3]{27 x^{10} y^{13}}\)[/tex]
We break this expression down in the same way:
[tex]\[ \sqrt[3]{27 x^{10} y^{13}} = \sqrt[3]{27} \cdot \sqrt[3]{x^{10}} \cdot \sqrt[3]{y^{13}} \][/tex]
- [tex]\(\sqrt[3]{27} = 3\)[/tex] because [tex]\(3^3 = 27\)[/tex]
- [tex]\(\sqrt[3]{x^{10}} = x^{3 + \frac{1}{3}}\)[/tex]
- [tex]\(\sqrt[3]{y^{13}} = y^{\frac{13}{3}}\)[/tex]
So,
[tex]\[ \sqrt[3]{27 x^{10} y^{13}} = 3 x^{3 + \frac{1}{3}} y^{\frac{13}{3}} \][/tex]
### Step 2: Simplify the remaining terms
3. [tex]\(8 x^3 y^4 (\sqrt[3]{x y})\)[/tex]
[tex]\(\sqrt[3]{x y} = x^{\frac{1}{3}} y^{\frac{1}{3}}\)[/tex], so:
[tex]\[ 8 x^3 y^4 (x^{\frac{1}{3}} y^{\frac{1}{3}}) = 8 x^{3+\frac{1}{3}} y^{4+\frac{1}{3}} = 8 x^{\frac{10}{3}} y^{\frac{13}{3}} \][/tex]
4. [tex]\(15 x^6 y^8 (\sqrt[3]{x y})\)[/tex]
Again, using [tex]\(\sqrt[3]{x y} = x^{\frac{1}{3}} y^{\frac{1}{3}}\)[/tex]:
[tex]\[ 15 x^6 y^8 (x^{\frac{1}{3}} y^{\frac{1}{3}}) = 15 x^{6+\frac{1}{3}} y^{8+\frac{1}{3}} = 15 x^{\frac{19}{3}} y^{\frac{25}{3}} \][/tex]
5. [tex]\(15 x^3 y^4 (\sqrt[3]{x y})\)[/tex]
As previously shown:
[tex]\[ 15 x^3 y^4 (x^{\frac{1}{3}} y^{\frac{1}{3}}) = 15 x^{\frac{10}{3}} y^{\frac{13}{3}} \][/tex]
6. [tex]\(8 x^6 y^8 (\sqrt[3]{x y})\)[/tex]
As previously shown:
[tex]\[ 8 x^6 y^8 (x^{\frac{1}{3}} y^{\frac{1}{3}}) = 8 x^{\frac{19}{3}} y^{\frac{25}{3}} \][/tex]
### Step 3: Combine like terms
Let's combine all terms that have the same exponents:
1.
[tex]\[ 5 x^{3 + \frac{1}{3}} y^{\frac{13}{3}} + 3 x^{3 + \frac{1}{3}} y^{\frac{13}{3}} + 8 x^{\frac{10}{3}} y^{\frac{13}{3}} + 15 x^{\frac{10}{3}} y^{\frac{13}{3}} \][/tex]
All these four terms share the factor [tex]\(x^{\frac{10}{3}} y^{\frac{13}{3}}\)[/tex].
So we combine these:
[tex]\[ (5 + 3 + 8 + 15) x^{\frac{10}{3}} y^{\frac{13}{3}} = 31 x^{\frac{10}{3}} y^{\frac{13}{3}} \][/tex]
2.
[tex]\[ 15 x^{\frac{19}{3}} y^{\frac{25}{3}} + 8 x^{\frac{19}{3}} y^{\frac{25}{3}} \][/tex]
Combine these using the same exponent property:
[tex]\[ (15 + 8) x^{\frac{19}{3}} y^{\frac{25}{3}} = 23 x^{\frac{19}{3}} y^{\frac{25}{3}} \][/tex]
### Final Expression
Putting it together, the total sum is:
[tex]\[ 31 x^{\frac{10}{3}} y^{\frac{13}{3}} + 23 x^{\frac{19}{3}} y^{\frac{25}{3}} \][/tex]
Thus, the sum of the given expressions is:
[tex]\(\boxed{31 x^{\frac{10}{3}} y^{\frac{13}{3}} + 23 x^{\frac{19}{3}} y^{\frac{25}{3}}}\)[/tex]
### Step 1: Simplify each cubic root term
1. [tex]\(\sqrt[3]{125 x^{10} y^{13}}\)[/tex]
We can break this expression down as follows:
[tex]\[ \sqrt[3]{125 x^{10} y^{13}} = \sqrt[3]{125} \cdot \sqrt[3]{x^{10}} \cdot \sqrt[3]{y^{13}} \][/tex]
- [tex]\(\sqrt[3]{125} = 5\)[/tex] because [tex]\(5^3 = 125\)[/tex]
- [tex]\(\sqrt[3]{x^{10}} = x^{\frac{10}{3}} = x^{3 + \frac{1}{3}}\)[/tex]
- [tex]\(\sqrt[3]{y^{13}} = y^{\frac{13}{3}}\)[/tex]
So,
[tex]\[ \sqrt[3]{125 x^{10} y^{13}} = 5 x^{3 + \frac{1}{3}} y^{\frac{13}{3}} \][/tex]
2. [tex]\(\sqrt[3]{27 x^{10} y^{13}}\)[/tex]
We break this expression down in the same way:
[tex]\[ \sqrt[3]{27 x^{10} y^{13}} = \sqrt[3]{27} \cdot \sqrt[3]{x^{10}} \cdot \sqrt[3]{y^{13}} \][/tex]
- [tex]\(\sqrt[3]{27} = 3\)[/tex] because [tex]\(3^3 = 27\)[/tex]
- [tex]\(\sqrt[3]{x^{10}} = x^{3 + \frac{1}{3}}\)[/tex]
- [tex]\(\sqrt[3]{y^{13}} = y^{\frac{13}{3}}\)[/tex]
So,
[tex]\[ \sqrt[3]{27 x^{10} y^{13}} = 3 x^{3 + \frac{1}{3}} y^{\frac{13}{3}} \][/tex]
### Step 2: Simplify the remaining terms
3. [tex]\(8 x^3 y^4 (\sqrt[3]{x y})\)[/tex]
[tex]\(\sqrt[3]{x y} = x^{\frac{1}{3}} y^{\frac{1}{3}}\)[/tex], so:
[tex]\[ 8 x^3 y^4 (x^{\frac{1}{3}} y^{\frac{1}{3}}) = 8 x^{3+\frac{1}{3}} y^{4+\frac{1}{3}} = 8 x^{\frac{10}{3}} y^{\frac{13}{3}} \][/tex]
4. [tex]\(15 x^6 y^8 (\sqrt[3]{x y})\)[/tex]
Again, using [tex]\(\sqrt[3]{x y} = x^{\frac{1}{3}} y^{\frac{1}{3}}\)[/tex]:
[tex]\[ 15 x^6 y^8 (x^{\frac{1}{3}} y^{\frac{1}{3}}) = 15 x^{6+\frac{1}{3}} y^{8+\frac{1}{3}} = 15 x^{\frac{19}{3}} y^{\frac{25}{3}} \][/tex]
5. [tex]\(15 x^3 y^4 (\sqrt[3]{x y})\)[/tex]
As previously shown:
[tex]\[ 15 x^3 y^4 (x^{\frac{1}{3}} y^{\frac{1}{3}}) = 15 x^{\frac{10}{3}} y^{\frac{13}{3}} \][/tex]
6. [tex]\(8 x^6 y^8 (\sqrt[3]{x y})\)[/tex]
As previously shown:
[tex]\[ 8 x^6 y^8 (x^{\frac{1}{3}} y^{\frac{1}{3}}) = 8 x^{\frac{19}{3}} y^{\frac{25}{3}} \][/tex]
### Step 3: Combine like terms
Let's combine all terms that have the same exponents:
1.
[tex]\[ 5 x^{3 + \frac{1}{3}} y^{\frac{13}{3}} + 3 x^{3 + \frac{1}{3}} y^{\frac{13}{3}} + 8 x^{\frac{10}{3}} y^{\frac{13}{3}} + 15 x^{\frac{10}{3}} y^{\frac{13}{3}} \][/tex]
All these four terms share the factor [tex]\(x^{\frac{10}{3}} y^{\frac{13}{3}}\)[/tex].
So we combine these:
[tex]\[ (5 + 3 + 8 + 15) x^{\frac{10}{3}} y^{\frac{13}{3}} = 31 x^{\frac{10}{3}} y^{\frac{13}{3}} \][/tex]
2.
[tex]\[ 15 x^{\frac{19}{3}} y^{\frac{25}{3}} + 8 x^{\frac{19}{3}} y^{\frac{25}{3}} \][/tex]
Combine these using the same exponent property:
[tex]\[ (15 + 8) x^{\frac{19}{3}} y^{\frac{25}{3}} = 23 x^{\frac{19}{3}} y^{\frac{25}{3}} \][/tex]
### Final Expression
Putting it together, the total sum is:
[tex]\[ 31 x^{\frac{10}{3}} y^{\frac{13}{3}} + 23 x^{\frac{19}{3}} y^{\frac{25}{3}} \][/tex]
Thus, the sum of the given expressions is:
[tex]\(\boxed{31 x^{\frac{10}{3}} y^{\frac{13}{3}} + 23 x^{\frac{19}{3}} y^{\frac{25}{3}}}\)[/tex]
