Answer :
To simplify the given expressions, we will combine like terms step-by-step.
1. Expression 1: [tex]\(2x^3 + 3y^3 + 5x^3 + 4y\)[/tex]
- Combine the [tex]\(x^3\)[/tex] terms: [tex]\(2x^3\)[/tex] and [tex]\(5x^3\)[/tex] add up to [tex]\(7x^3\)[/tex].
- The [tex]\(y^3\)[/tex] term is [tex]\(3y^3\)[/tex].
- The [tex]\(y\)[/tex] term is [tex]\(4y\)[/tex].
- Simplified form: [tex]\(7x^3 + 3y^3 + 4y\)[/tex].
2. Expression 2: [tex]\(7x^6 + 7y^4\)[/tex]
- No like terms to combine here.
- Simplified as is: [tex]\(7x^6 + 7y^4\)[/tex].
3. Expression 3: [tex]\(7x^3 + 3y^3 + 4y\)[/tex]
- Already simplified with terms grouped by powers.
- Simplified as is: [tex]\(7x^3 + 3y^3 + 4y\)[/tex].
4. Expression 4: [tex]\(7x^6 + 3y^3 + 4y\)[/tex]
- Segregated into different terms:
- [tex]\(7x^6\)[/tex] for the [tex]\(x^6\)[/tex] term.
- [tex]\(3y^3\)[/tex] for the [tex]\(y^3\)[/tex] term.
- [tex]\(4y\)[/tex] for the [tex]\(y\)[/tex] term.
- Simplified as is: [tex]\(7x^6 + 3y^3 + 4y\)[/tex].
5. Expression 5: [tex]\(5xy^3 + 5x^3 + 4y\)[/tex]
- Terms identified as:
- [tex]\(5xy^3\)[/tex] for the mixed term.
- [tex]\(5x^3\)[/tex] for the [tex]\(x^3\)[/tex] term.
- [tex]\(4y\)[/tex] for the [tex]\(y\)[/tex] term.
- Simplified as is: [tex]\(5xy^3 + 5x^3 + 4y\)[/tex].
Now, combining all expressions together:
- Terms for [tex]\(x^6\)[/tex]: From Expression 2 and 4, we have a total of [tex]\(7x^6\)[/tex].
- Terms for [tex]\(x^3\)[/tex]: Combine from Expression 1 [tex]\((7x^3)\)[/tex] and Expression 5 [tex]\((5x^3)\)[/tex] to get [tex]\(12x^3\)[/tex].
- Terms for [tex]\(xy^3\)[/tex]: From Expression 5, we have [tex]\(5xy^3\)[/tex].
- Terms for [tex]\(y^4\)[/tex]: From Expression 2, we have [tex]\(7y^4\)[/tex].
- Terms for [tex]\(y^3\)[/tex]: Combine from Expression 1 [tex]\((3y^3)\)[/tex] and Expression 3 [tex]\((3y^3)\)[/tex] giving a total of [tex]\(6y^3\)[/tex].
- Terms for [tex]\(y\)[/tex]: Combine from Expression 1 [tex]\((4y)\)[/tex], Expression 3 [tex]\((4y)\)[/tex], and Expression 5 [tex]\((4y)\)[/tex].
The simplified expression in total gives us:
- [tex]\(x^6: 7\)[/tex]
- [tex]\(x^3: 12\)[/tex]
- [tex]\(xy^3: 5\)[/tex]
- [tex]\(y^4: 7\)[/tex]
- [tex]\(y^3: 6\)[/tex]
- [tex]\(y: 4\)[/tex]
The final simplified result is:
[tex]\[
7x^6 + 12x^3 + 5xy^3 + 7y^4 + 6y^3 + 4y
\][/tex]
1. Expression 1: [tex]\(2x^3 + 3y^3 + 5x^3 + 4y\)[/tex]
- Combine the [tex]\(x^3\)[/tex] terms: [tex]\(2x^3\)[/tex] and [tex]\(5x^3\)[/tex] add up to [tex]\(7x^3\)[/tex].
- The [tex]\(y^3\)[/tex] term is [tex]\(3y^3\)[/tex].
- The [tex]\(y\)[/tex] term is [tex]\(4y\)[/tex].
- Simplified form: [tex]\(7x^3 + 3y^3 + 4y\)[/tex].
2. Expression 2: [tex]\(7x^6 + 7y^4\)[/tex]
- No like terms to combine here.
- Simplified as is: [tex]\(7x^6 + 7y^4\)[/tex].
3. Expression 3: [tex]\(7x^3 + 3y^3 + 4y\)[/tex]
- Already simplified with terms grouped by powers.
- Simplified as is: [tex]\(7x^3 + 3y^3 + 4y\)[/tex].
4. Expression 4: [tex]\(7x^6 + 3y^3 + 4y\)[/tex]
- Segregated into different terms:
- [tex]\(7x^6\)[/tex] for the [tex]\(x^6\)[/tex] term.
- [tex]\(3y^3\)[/tex] for the [tex]\(y^3\)[/tex] term.
- [tex]\(4y\)[/tex] for the [tex]\(y\)[/tex] term.
- Simplified as is: [tex]\(7x^6 + 3y^3 + 4y\)[/tex].
5. Expression 5: [tex]\(5xy^3 + 5x^3 + 4y\)[/tex]
- Terms identified as:
- [tex]\(5xy^3\)[/tex] for the mixed term.
- [tex]\(5x^3\)[/tex] for the [tex]\(x^3\)[/tex] term.
- [tex]\(4y\)[/tex] for the [tex]\(y\)[/tex] term.
- Simplified as is: [tex]\(5xy^3 + 5x^3 + 4y\)[/tex].
Now, combining all expressions together:
- Terms for [tex]\(x^6\)[/tex]: From Expression 2 and 4, we have a total of [tex]\(7x^6\)[/tex].
- Terms for [tex]\(x^3\)[/tex]: Combine from Expression 1 [tex]\((7x^3)\)[/tex] and Expression 5 [tex]\((5x^3)\)[/tex] to get [tex]\(12x^3\)[/tex].
- Terms for [tex]\(xy^3\)[/tex]: From Expression 5, we have [tex]\(5xy^3\)[/tex].
- Terms for [tex]\(y^4\)[/tex]: From Expression 2, we have [tex]\(7y^4\)[/tex].
- Terms for [tex]\(y^3\)[/tex]: Combine from Expression 1 [tex]\((3y^3)\)[/tex] and Expression 3 [tex]\((3y^3)\)[/tex] giving a total of [tex]\(6y^3\)[/tex].
- Terms for [tex]\(y\)[/tex]: Combine from Expression 1 [tex]\((4y)\)[/tex], Expression 3 [tex]\((4y)\)[/tex], and Expression 5 [tex]\((4y)\)[/tex].
The simplified expression in total gives us:
- [tex]\(x^6: 7\)[/tex]
- [tex]\(x^3: 12\)[/tex]
- [tex]\(xy^3: 5\)[/tex]
- [tex]\(y^4: 7\)[/tex]
- [tex]\(y^3: 6\)[/tex]
- [tex]\(y: 4\)[/tex]
The final simplified result is:
[tex]\[
7x^6 + 12x^3 + 5xy^3 + 7y^4 + 6y^3 + 4y
\][/tex]
