MHB Adding numbers with exponents (confusion)

AI Thread Summary
The discussion clarifies the confusion between adding terms with exponents and coefficients. It explains that while 3x^2 + 4x^2 can be simplified to 7x^2 using the distributive property, the same cannot be applied to 3^2 + 4^2, which equals 25. The key difference lies in the structure of the expressions, as the distributive law does not apply to constants alone. By comparing similar expressions, participants gain a clearer understanding of how coefficients and bases operate differently. This insight helps solidify the concept that coefficients can be added directly while exponents must be calculated separately.
some one1
Messages
2
Reaction score
0
Alright here's my confusion, if i take say 3x^2 + 4x^2 ill end up with 7x^2 which i accepted was the correct way to think about it, but if i try the same problem without the x variable doing the same method, 3^2 + 4^2 = 7^2 this is obviously not the correct answer. Instead 3^2 = 9 and 4^2 = 16 so together they equal 25 (7^2 = 49 is incorrect!)

Now if your planning on telling me i should just treat 3x^2 differently then 3^2 without explaining why, well that is not going to help my understanding. I need someone to explain it to me using a common sense approach as to why you do this instead of this, just following rules blindly is next to magic when it comes to trying to fully understand what's going on.

I really appreciate the help, math has always been my weak point.
 
Mathematics news on Phys.org
$$3x^2=x^2+x^2+x^2$$

$$4x^2=x^2+x^2+x^2+x^2$$

$$3x^2+4x^2=x^2+x^2+x^2+x^2+x^2+x^2+x^2=7x^2$$
 
In rewriting $3x^2+4x^2$ to $7x^2$ we use the law of distributivity of multiplication over addition. This law says
\[
(a+b)c=ac+bc\qquad(1)
\]
for all numbers $a$, $b$ and $c$. In this case, $a=3$, $b=4$ and $c=x^2$. Substituting these values into (1) gives
\[
(3+4)x^2=3x^2+4x^2
\]
so we can indeed rewrite the right-hand side to the left-hand side and then rewrite it further to $7x^2$ since $3+4=7$.

On the other hand, the expression $3^2+4^2$ simply does not have the shape of either the left- or the right-hand side of (1). You can't match it with (1), i.e., you can't come up with three values such that replacing $a$, $b$ and $c$ in (1) by those values would give $3^2+4^2$. Therefore, (1) can't be used to rewrite $3^2+4^2$.
 
greg1313 said:
$$3x^2=x^2+x^2+x^2$$

$$4x^2=x^2+x^2+x^2+x^2$$

$$3x^2+4x^2=x^2+x^2+x^2+x^2+x^2+x^2+x^2=7x^2$$

Thanks, i think i understand what i was doing wrong with how i was looking at it, for example, 2x^2 + 2x^2 = 4x^2, if x=2 then 4x^2 = 16 which is the same as 2^3+2^3 which equals 16.
looking at how they are the same helps me to see the obvious mistake i was making, i was looking at 2^3 and was confusing the base with how i understood coefficients to work. but by comparing them with equal examples to one another i saw the obvious difference and mistake in my understanding.

Thank you for helping me, it has finally clicked with my common sense.
 
$3x^2+4x^2$ is the same as 3 apples + 4 apples...
 
Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...
Suppose ,instead of the usual x,y coordinate system with an I basis vector along the x -axis and a corresponding j basis vector along the y-axis we instead have a different pair of basis vectors ,call them e and f along their respective axes. I have seen that this is an important subject in maths My question is what physical applications does such a model apply to? I am asking here because I have devoted quite a lot of time in the past to understanding convectors and the dual...

Similar threads

Replies
10
Views
2K
Replies
3
Views
1K
Replies
5
Views
2K
Replies
12
Views
2K
Replies
2
Views
1K
Replies
10
Views
2K
Back
Top