Simplifying Expressions with Multiple Variables

A power records how many equal factors are being multiplied. Here x is used as a factor 3 times, so

$x\times x\times x=x^3$.

Two checks worth making: the index counts the x's (three), not the multiplication signs (two); and multiplying is not the same as adding: $x+x+x$ would be $3x$, which is different from $x^3$.

$4x$ and $3x$ are like terms (both are a number of x), so they can be collected into one term.

Add the coefficients and keep the same letter: $4+3=7$, so $4x+3x=7x$.

The coefficients are added, not multiplied, the letter stays a single x (adding x terms never produces $x^2$), and terms with different coefficients are still like terms.

No. Like terms must contain exactly the same letters raised to exactly the same powers; only the coefficients may differ.

In $2a$ the letter a has index 1. In $a^2$ it has index 2. The powers are different, so the terms are not like terms and $2a+a^2$ cannot be collected into a single term.

Note also that $2a$ means $a+a$, while $a^2$ means $a\times a$. These are only equal when $a=0$ or $a=2$.

Collect the terms in a and the terms in b separately, keeping each sign with its term.

a terms: $2a+5a=7a$.

b terms: $3b-b$. Remember $b$ means $1b$, so $3b-1b=2b$.

Different coefficients do not stop terms being like terms, and a and b are different letters, so $7a$ and $2b$ cannot be combined any further.

The simplified expression is $7a+2b$.

No. Like terms must have exactly the same letters raised to exactly the same powers.

Written out, $4a{b}^2=4\times a\times b\times b$ and $3a^{2}b=3\times a\times a\times b$. Both use a and b, but the powers are swapped: the first has $a^1{b}^2$ and the second has $a^2{b}^1$.

Because the powers do not match, the terms are not like terms and cannot be added together.

Deal with each type of term separately.

$x^2$ terms: $7x^2-2x^2=5x^2$.

$xy$ terms: multiplication can be done in any order, so $yx$ is the same as $xy$, and $-yx$ means $-1xy$. So $3xy-xy=2xy$.

$5x^2$ and $2xy$ are not like terms (one is $x\times x$, the other $x\times y$), so they stay apart.

The simplified expression is $5x^2+2xy$.