Work the operations in this order: Square first, multiply second, add third.

$\displaystyle 6 \times 3 + 4 ^{2} = 6 \times 3 +\left ( 4 \times 4 \right ) = 6 \times 3 + 16 = 18 + 16 = 34$

When solving this problem, you need to remember order of operations, which tells us which operation has to be done first.

When following order of operations, you complete operations in the following order:

1. Parenthesis - Complete any operation in parenthesis first

2. Exponents - Solve exponents second

3. Multiplication and Division - Multiply or Divide (whichever one comes first from left to right)

4. Addition and Subtraction - Add or Subtract (whichever one comes first from left to right)

Keeping order of operations in mind, the steps to solve $\displaystyle 3-4+2^{3}*3$ are below

$\displaystyle 3-4+2^{3}*3 \textup{ } (\textup{Solve for the exponent})$

$\displaystyle 3-4+8*3 \textup{ } (\textup{Multiply } 8 \textup{ and }3)$

$\displaystyle 3-4+24 \textup{ } (\textup{Subtract } 3 \textup{ and }4)$ 

$\displaystyle -1+24 \textup{ } (\textup{Add } -1 \textup{ and }24)$

$\displaystyle 23$

Substitute 6 for $\displaystyle a$ and $\displaystyle -6$ for $\displaystyle b$:

$\displaystyle a \bigstar b = \frac{6 + 5}{6 + (-6) + 5}= \frac{11}{0 + 5} = \frac{11}{5} = 2\frac{1}{5}$

Substitute 7 for $\displaystyle a$ and -$\displaystyle 12$ for $\displaystyle b$:

$\displaystyle a \bigstar b = \frac{a + 5}{a + b + 5}$

$\displaystyle 7 \bigstar -12 = \frac{7 + 5}{7 + (-12)+ 5} = \frac{12}{-5+ 5} = \frac{12}{0}$

Any fraction with a zero denominator - such as this one - is an undefined quantity.

This problem involves order of operations.

Use the acronym for the correct order:

PEMDAS-Parenthesis, Exponents, Multiply, Divide, Add, Subtract

(Please Excuse My Dear Aunt Sally)

Start by grouping what needs to be solved first.

$\displaystyle -2 + 3\times6-12\div3+2$

$\displaystyle -2 +\left ( 3\times6 \right )-\left (12\div3 \right )+2$

$\displaystyle -2+18-4+2$

The answer is 14.

By the order of operations, in the absence of grouping symbols, exponents must be carried out before any other operations. The correct choice is that eight must be squared.

Substitute 0.5 for $\displaystyle y$ and follow the order of operations:

$\displaystyle y ^{2} - 6y + 7$

$\displaystyle =0.5 ^{2} - 6(0.5) + 7$

$\displaystyle =0.25- 6(0.5) + 7$

$\displaystyle =0.25 - 3 + 7$

$\displaystyle = -2.75 + 7$

$\displaystyle = 4.25$

Substitute $\displaystyle - 6$ for $\displaystyle x$ in the expression and folllow the order of operations:

$\displaystyle -7x + 13$

$\displaystyle = -7 (-6) + 13$

$\displaystyle = +(7 \cdot 6)+ 13$

$\displaystyle = 42 + 13$

$\displaystyle = 55$

Substitute 7 for $\displaystyle x$ in the expression and evaluate, paying attention to the order of operations:

$\displaystyle -7 + x^{2}$

$\displaystyle = -7 + 7^{2}$

$\displaystyle = -7 + 49$

$\displaystyle = +\left ( 49 - 7 \right )$

$\displaystyle =42$

In the order of operations, additions and subtractions are carried out from left to right:

$\displaystyle 800 - 281 + 154 - 165 + 342$

$\displaystyle =519 + 154 - 165 + 342$

$\displaystyle =673 - 165 + 342$

$\displaystyle =508 + 342$

$\displaystyle = 850$