Answer :
- Add the expressions: $(3x^4 + 2x^4) + (4x^4 + 7x^3)$.
- Combine like terms: $(3+2+4)x^4 + 7x^3$.
- Simplify: $9x^4 + 7x^3$.
- The correct answer is $9x^4 + 7x^3$, but none of the options match. There might be a typo in the options or the question.
### Explanation
1. Understanding the problem
We are given two expressions, $3x^4 + 2x^4$ and $4x^4 + 7x^3$, and we want to find their sum.
2. Adding the expressions
To find the sum, we add the two expressions together: $(3x^4 + 2x^4) + (4x^4 + 7x^3)$.
3. Combining like terms
Now, we combine like terms. Like terms are terms that have the same variable raised to the same power. In this case, we have $3x^4$, $2x^4$, and $4x^4$ as like terms. So, we add their coefficients: $3 + 2 + 4 = 9$. This gives us $9x^4$. The term $7x^3$ does not have any like terms, so it remains as it is.
4. Simplifying the expression
Therefore, the sum is $9x^4 + 7x^3$.
5. Finding the equivalent expression
Comparing our simplified expression $9x^4 + 7x^3$ with the given options, we see that it matches option D if we rearrange the terms. However, none of the options exactly match our result. Let's re-examine our steps to make sure we didn't make a mistake. We have $(3x^4 + 2x^4) + (4x^4 + 7x^3) = (3+2+4)x^4 + 7x^3 = 9x^4 + 7x^3$. The options are:
A) $16 x^4$
B) $7 x^8+9 x^6$
C) $12 x^4+14 x^3$
D) $7 x^4+9 x^3$
It seems there was a typo in the question or the options. The correct sum is $9x^4 + 7x^3$, but none of the options match this. However, if the question was asking for the sum of $3x^4 + 4x^4$ and $2x^4 + 7x^3$, then the sum would be $(3x^4 + 4x^4) + (2x^4 + 7x^3) = (3+4+2)x^4 + 7x^3 = 9x^4 + 7x^3$. Still no match.
If the question was the sum of $3x^4 + 2x^4$ and $4x^4 + 7x^3$, and option D was $9x^4 + 7x^3$, then D would be correct. Given the options, let's assume the question intended to ask for the sum of $3x^4 + 2x^4$ and $4x^4 + 7x^3$, and option D is the closest to the correct answer if we swap the coefficients, but it's still incorrect.
6. Final Answer
The correct sum of the given expressions is $9x^4 + 7x^3$. However, since none of the options match this result, and assuming there might be a typo in the options, we will proceed by finding the closest option. Option D, $7x^4 + 9x^3$, has the correct terms but with swapped coefficients. Therefore, none of the provided options are equivalent to the sum of $3 x^4+2 x^4$ and $4 x^4+7 x^3$.
### Examples
Understanding how to combine like terms is essential in many areas of mathematics and science. For example, when calculating the total area of a garden that is divided into different sections, you need to combine the areas of each section, which often involves adding expressions with like terms. Similarly, in physics, when analyzing forces acting on an object, you need to combine force vectors, which can involve adding expressions with like components.
- Combine like terms: $(3+2+4)x^4 + 7x^3$.
- Simplify: $9x^4 + 7x^3$.
- The correct answer is $9x^4 + 7x^3$, but none of the options match. There might be a typo in the options or the question.
### Explanation
1. Understanding the problem
We are given two expressions, $3x^4 + 2x^4$ and $4x^4 + 7x^3$, and we want to find their sum.
2. Adding the expressions
To find the sum, we add the two expressions together: $(3x^4 + 2x^4) + (4x^4 + 7x^3)$.
3. Combining like terms
Now, we combine like terms. Like terms are terms that have the same variable raised to the same power. In this case, we have $3x^4$, $2x^4$, and $4x^4$ as like terms. So, we add their coefficients: $3 + 2 + 4 = 9$. This gives us $9x^4$. The term $7x^3$ does not have any like terms, so it remains as it is.
4. Simplifying the expression
Therefore, the sum is $9x^4 + 7x^3$.
5. Finding the equivalent expression
Comparing our simplified expression $9x^4 + 7x^3$ with the given options, we see that it matches option D if we rearrange the terms. However, none of the options exactly match our result. Let's re-examine our steps to make sure we didn't make a mistake. We have $(3x^4 + 2x^4) + (4x^4 + 7x^3) = (3+2+4)x^4 + 7x^3 = 9x^4 + 7x^3$. The options are:
A) $16 x^4$
B) $7 x^8+9 x^6$
C) $12 x^4+14 x^3$
D) $7 x^4+9 x^3$
It seems there was a typo in the question or the options. The correct sum is $9x^4 + 7x^3$, but none of the options match this. However, if the question was asking for the sum of $3x^4 + 4x^4$ and $2x^4 + 7x^3$, then the sum would be $(3x^4 + 4x^4) + (2x^4 + 7x^3) = (3+4+2)x^4 + 7x^3 = 9x^4 + 7x^3$. Still no match.
If the question was the sum of $3x^4 + 2x^4$ and $4x^4 + 7x^3$, and option D was $9x^4 + 7x^3$, then D would be correct. Given the options, let's assume the question intended to ask for the sum of $3x^4 + 2x^4$ and $4x^4 + 7x^3$, and option D is the closest to the correct answer if we swap the coefficients, but it's still incorrect.
6. Final Answer
The correct sum of the given expressions is $9x^4 + 7x^3$. However, since none of the options match this result, and assuming there might be a typo in the options, we will proceed by finding the closest option. Option D, $7x^4 + 9x^3$, has the correct terms but with swapped coefficients. Therefore, none of the provided options are equivalent to the sum of $3 x^4+2 x^4$ and $4 x^4+7 x^3$.
### Examples
Understanding how to combine like terms is essential in many areas of mathematics and science. For example, when calculating the total area of a garden that is divided into different sections, you need to combine the areas of each section, which often involves adding expressions with like terms. Similarly, in physics, when analyzing forces acting on an object, you need to combine force vectors, which can involve adding expressions with like components.
