By the order of operations, in the absence of grouping symbols, multiplication takes precedence over addition and subtraction. Addition and subtraction are then carried out with equal priority, but from left to right, so the addition is performed second and the subtraction last.

In modulo 8 arithmetic, a number is congruent to the remainder of the divison of that number by 8. Since

$\displaystyle 6 \times 3 \times 2 = 36$

and 

$\displaystyle 36 \div 8 = 4 \textrm{ R }4$

then 

$\displaystyle 6 \times 3 \times 2 \equiv 4 \mod 8$.

The correct response is 4.

$\displaystyle 9x - 55$ is fifty-five subtracted from $\displaystyle 9x$.

$\displaystyle 9x$ is the product of nine and a number.

Subsequently, $\displaystyle 9x - 55$ is "fifty-five subtracted from the product of nine and a number".

When multiplying like variables, the constants are multiplied together.  

For the exponents, when you multply variable exponents you have to add the exponents together.  

$\displaystyle 5*2=10$ and $\displaystyle 2+3=5$ so that gives you an answer of $\displaystyle 10x^{5}$.

When multiplying variables with exponents, add the exponents.

$\displaystyle (x^{4 })(x^{2}) = x^{4+2} = x^{6}$

$\displaystyle (y^{5})(y^{1}) =y ^{5+1} =y^{6}$

$\displaystyle z^{1} = z$

The correct answer is $\displaystyle x^{6}y^{6}z$

When multiplying variables with exponents the exponents are added.

$\displaystyle (a)(a) = a^{1+1} =a^{2}$

$\displaystyle (b^{3}) (b^{4})= b^{3+4} = b^{7}$

$\displaystyle (c^{4}) (c^{2}) = c^{4+2} = c^{6}$

The correct answer is $\displaystyle a^{2}b^{7}c^{6}$

She needs $\displaystyle 25+10+10+1+1$ to make $\displaystyle 47$ cents.

You can only add like terms. Therefore, different variables are treated as different types of terms. Since $\displaystyle 2x$ and $\displaystyle 3x$ both end in the variable $\displaystyle x$, they can be added together. The $\displaystyle 5b$ cannot be added to these numbers; however, because it has a different variable. The answer is:

$\displaystyle 5x + 5b$

The associative property of addition allows us to group the numbers with the same variables together: $\displaystyle 2x+3x+4y+6xy$

The like terms in this expression are:

• $\displaystyle 2x$ and $\displaystyle 3x$
• $\displaystyle 4y$
• $\displaystyle 6xy$

Terms with different variables cannot be grouped together.

As a result, the only way to simplify this expression is to add $\displaystyle 2x$ and $\displaystyle 3x$.

When adding variables of the same type, they are simply added together, and the variable remains to the first power. This is known as combining like-terms.

$\displaystyle 7x+2x+5x=9x+5x=14x$

Thus, the correct answer is $\displaystyle 14x$.