Help with adding and subtracting exponents -

Question

I have a problem in my mathematics (as math usually goes...) and it goes like this:

$Simplify.$

$(x^4-2x^2y-7)+(7x^4+6xy-3)$

So, I reordered the equation for simplicity's sake.

$x^4+7x^4-2x^2y+6xy-7-3$

Then, obvously, I simplified the equation and I concluded that the answer must be:

$8x^8+4x^3y^2-10$

But nope.. the answer is

$8x^8-2x^2y+6xy-10$

Why is this? I thought that you had to add the like terms, which in my mind, $2x^2y$ and $6xy$ where - clearly I was mistaken. Why dont you add the $x$ and $y$ for those two terms, but add the $x$ and $y$ variables in the other terms? This is bugging me as I can't clearly see/explain this.

Would someone mind explaining this to me quickly?
Thanks for your help!

Answer 1

In general, $x^2 \neq x$. If we multiply both sides by $y$ we get that $x^2 y \neq xy$. In general you can combine terms only when all variables are the same in each term and each variable is raised to the same power as all others of its type - only coefficients can vary. For example, take $5x^3y^2$. Right off the bat we know we can only combine this with another term $C x^3 y^2$, where $C$ is any constant value. When we do this we carry out the distributive property in reverse to get $(5+C)x^3y^2$. Note that the combined terms have the same variables and exponents on said variables as the answer we got... This is why it didn't work when you tried to combine unlike terms.

Answer 2

Hint:

Think at the properties of the operations that you are using.

E.g.:

by definition: $$ x^4+7x^4=1\cdot x^4+7\cdot x^4= $$ by distributivity of the product over the addition: $$ (1+7)\cdot x^4=8x^4 $$

and so one for the other terms... but, for

$$ -2x^2y+6xy $$

can you use some property ?