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Answer :
To simplify the given mathematical expression, we need to combine like terms. The expression you provided is as follows:
[tex]\[ 2x^3 + 3y^3 + 5x^3 + 4y + 5xy^3 + 5x^3 + 4y + 7x^6 + 3y^3 + 4y + 7x^3 + 3y^3 + 4y + 7x^6 + 7y^4 \][/tex]
Let's break it down step-by-step:
1. Identify Like Terms:
- Combine the terms with the same variables raised to the same powers.
2. Group Like Terms:
- Terms with [tex]\(x^6\)[/tex]: [tex]\(7x^6 + 7x^6 = 14x^6\)[/tex]
- Terms with [tex]\(x^3\)[/tex]: [tex]\(2x^3 + 5x^3 + 5x^3 + 7x^3 = 19x^3\)[/tex]
- Terms with [tex]\(xy^3\)[/tex]: [tex]\(5xy^3\)[/tex] (only one term)
- Terms with [tex]\(y^4\)[/tex]: [tex]\(7y^4\)[/tex] (only one term)
- Terms with [tex]\(y^3\)[/tex]: [tex]\(3y^3 + 3y^3 + 3y^3 = 9y^3\)[/tex]
- Constant [tex]\(y\)[/tex] terms: [tex]\(4y + 4y + 4y + 4y = 16y\)[/tex]
3. Construct the Simplified Expression:
- Combine all the grouped like terms into a single expression:
[tex]\[ 14x^6 + 19x^3 + 5xy^3 + 7y^4 + 9y^3 + 16y \][/tex]
This is the simplified form of the original expression, where all like terms have been combined correctly.
[tex]\[ 2x^3 + 3y^3 + 5x^3 + 4y + 5xy^3 + 5x^3 + 4y + 7x^6 + 3y^3 + 4y + 7x^3 + 3y^3 + 4y + 7x^6 + 7y^4 \][/tex]
Let's break it down step-by-step:
1. Identify Like Terms:
- Combine the terms with the same variables raised to the same powers.
2. Group Like Terms:
- Terms with [tex]\(x^6\)[/tex]: [tex]\(7x^6 + 7x^6 = 14x^6\)[/tex]
- Terms with [tex]\(x^3\)[/tex]: [tex]\(2x^3 + 5x^3 + 5x^3 + 7x^3 = 19x^3\)[/tex]
- Terms with [tex]\(xy^3\)[/tex]: [tex]\(5xy^3\)[/tex] (only one term)
- Terms with [tex]\(y^4\)[/tex]: [tex]\(7y^4\)[/tex] (only one term)
- Terms with [tex]\(y^3\)[/tex]: [tex]\(3y^3 + 3y^3 + 3y^3 = 9y^3\)[/tex]
- Constant [tex]\(y\)[/tex] terms: [tex]\(4y + 4y + 4y + 4y = 16y\)[/tex]
3. Construct the Simplified Expression:
- Combine all the grouped like terms into a single expression:
[tex]\[ 14x^6 + 19x^3 + 5xy^3 + 7y^4 + 9y^3 + 16y \][/tex]
This is the simplified form of the original expression, where all like terms have been combined correctly.
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