What is the exponent of the last term of $(2x^2+3y^3)^{10}$?

Question

What is the exponent of the last term of: $$(2x^2+3y^3)^{10}$$

Hi! I'm sorry if this question seems a bit amateurish. I'm quite confused with this question that was asked in a quiz about binomial expansion.

My teacher said that the answer is 30. However, isn't 30 the exponent of y and not the exponent of the whole last term? The b term would be: $$3y^3$$ and so the last term would be $$(3y^3)^{10}=3^{10}y^{30}$$

Isn't the answer suppose to be 10 since the exponent of: $$3y^3$$ is 10.

Yes it is the exponent of $y$ and I think the question also asks this. Last term is $3^{10}y^{30}$ and the exponent of this term is $30$. — Seyhmus Güngören, Aug 29 '12 at 10:54
The question asks for the exponent of the entire last term which is $$(3y^3)^{10}$$. If you factor out the exponents in $$3^{10}y^{30}$$, wouldn't the exponent of the entire term be 10? — Mico Abrina, Aug 29 '12 at 11:07
yes you are right. exponent of the entire term is $10$ but the last term of the binomial expension is $30$ — Seyhmus Güngören, Aug 29 '12 at 11:10
However, the question is asking for the exponent of the entire last term and not just y. Y is considered a term but I don't think it's the last term of the expansion. I think the problem lies with the question since the term - "last term" is quite ambiguous. — Mico Abrina, Aug 29 '12 at 11:30
please think how you like. You already have the solution in mind then no need to ask even. Better discuss with your teacher. — Seyhmus Güngören, Aug 29 '12 at 13:46

Marc van Leeuwen · Accepted Answer · 2012-08-29T11:44:35.037

4

Both the notions "exponent of a term" and "last term" are ill defined. Terms can be ordered arbitrarily, and can be expanded, thereby changing their composition without loosing their identity (i.e., most people would consider it to still be the same term, although it looks different). However if you ask for the exponent in the term $(3y^3)^{10}$ after expansion, then the answer is $30$, since expansion makes that term $59049y^{30}$ which contains only one exponent, $30$. Before expansion it contain two exponents, $3$ and $10$. And strictly speaking a term never has an exponent (attached to it) since a term is by definition an operand of an additive operation, not of exponentiation, so any exponent is part of the term.

edited Aug 29 '12 at 11:44

answered Aug 29 '12 at 11:39

Marc van Leeuwen

115,048

Does that mean then that the problem is due to the question being too ambiguous? – Mico Abrina Aug 29 '12 at 11:45
1

Definitely$ $ $ $ – Marc van Leeuwen Aug 29 '12 at 11:46
Therefore, the question is invalid. I'll be talking to my teacher about this. I would give your answer a check and a vote but unfortunately I don't have enough points to do that. – Mico Abrina Aug 29 '12 at 11:49
@MicoAbrina: I don't think you need any reputation to accept answers, or even to vote-up answers, certainly to you own question. – Marc van Leeuwen Aug 29 '12 at 11:54
Good point. I gave it a vote and I marked it as the right answer. – Mico Abrina Aug 29 '12 at 12:00

score 0 · Answer 2 · answered Aug 29 '12 at 10:54

0

I think your teacher meant the exponent of the variable $y$, and not of the entire term. The number $3^{10}$ would be called the coefficient of the exponent $y^{30}$.

answered Aug 29 '12 at 10:54

Nathaniel Bubis

33,018

In the test, the question was specifically asking for the exponent of the last term and not y. – Mico Abrina Aug 29 '12 at 11:03
@MicoAbrina - sounds like you should talk to your teacher.. – Nathaniel Bubis Aug 29 '12 at 11:11

What is the exponent of the last term of $(2x^2+3y^3)^{10}$?

2 Answers2