Adding numbers with exponents (confusion)

[some one1]

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.

 

[Greg]

\(\displaystyle 3x^2=x^2+x^2+x^2\)

\(\displaystyle 4x^2=x^2+x^2+x^2+x^2\)

\(\displaystyle 3x^2+4x^2=x^2+x^2+x^2+x^2+x^2+x^2+x^2=7x^2\)

 

[Evgeny.Makarov]

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 $3^2+4^2$.

 

[some one1]

\(\displaystyle 3x^2=x^2+x^2+x^2\)

\(\displaystyle 4x^2=x^2+x^2+x^2+x^2\)

\(\displaystyle 3x^2+4x^2=x^2+x^2+x^2+x^2+x^2+x^2+x^2=7x^2\)

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.

 

[Wilmer]

$3x^2+4x^2$ is the same as 3 apples + 4 apples...