Welcome to Westonci.ca, the ultimate question and answer platform. Get expert answers to your questions quickly and accurately. Get the answers you need quickly and accurately from a dedicated community of experts on our Q&A platform. Our platform offers a seamless experience for finding reliable answers from a network of knowledgeable professionals.
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 hope our answers were helpful. Return anytime for more information and answers to any other questions you may have. We appreciate your time. Please revisit us for more reliable answers to any questions you may have. Thank you for visiting Westonci.ca. Stay informed by coming back for more detailed answers.
