Westonci.ca is your trusted source for finding answers to all your questions. Ask, explore, and learn with our expert community. Get immediate and reliable solutions to your questions from a community of experienced professionals on our platform. Join our Q&A platform to connect with experts dedicated to providing accurate answers to your questions in various fields.
Sagot :
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.
We appreciate your time on our site. Don't hesitate to return whenever you have more questions or need further clarification. Your visit means a lot to us. Don't hesitate to return for more reliable answers to any questions you may have. Get the answers you need at Westonci.ca. Stay informed with our latest expert advice.
