To determine whether [tex]\(4xy^3\)[/tex] and [tex]\(-5x^3y\)[/tex] are like terms, we need to look closely at both their variables and the powers to which those variables are raised.
Let's break down the terms:
1. [tex]\(4xy^3\)[/tex]:
- Coefficient: 4
- Variables and their exponents: [tex]\(x\)[/tex] to the power of 1 and [tex]\(y\)[/tex] to the power of 3
2. [tex]\(-5x^3y\)[/tex]:
- Coefficient: -5
- Variables and their exponents: [tex]\(x\)[/tex] to the power of 3 and [tex]\(y\)[/tex] to the power of 1
For two terms to be like terms, their variables must be identical with exactly the same exponents. Let's compare the exponents of the variables in the given terms:
- For the first term [tex]\(4xy^3\)[/tex]:
- [tex]\(x\)[/tex] has an exponent of 1
- [tex]\(y\)[/tex] has an exponent of 3
- For the second term [tex]\(-5x^3y\)[/tex]:
- [tex]\(x\)[/tex] has an exponent of 3
- [tex]\(y\)[/tex] has an exponent of 1
We see that:
- The exponent of [tex]\(x\)[/tex] in [tex]\(4xy^3\)[/tex] is 1, while in [tex]\(-5x^3y\)[/tex] it is 3.
- The exponent of [tex]\(y\)[/tex] in [tex]\(4xy^3\)[/tex] is 3, while in [tex]\(-5x^3y\)[/tex] it is 1.
Since the exponents of the variables in the two terms are not identical (i.e., [tex]\(x\)[/tex] is raised to different powers, and [tex]\(y\)[/tex] is raised to different powers), the terms are not like terms. The correct answer must explain that the difference in the exponents of the variables means Lena is incorrect in her statement.
Hence, the correct answer is:
D. No, because the same variables are not raised to the same powers.
