Arithmetic Operator Precedence

The order of arithmetic operator precedence is as follows:

1. Parenthesis (expressions within brackets such as,(), [] etc.)
2. Exponents (powers like \(2^2\))
3. Division (\(a \div b\)) & Multiplication (\(a \times b)\)
4. Addition (\(a + b\)) & subtraction (\(a – b\))

Operations with equal precedence (addition and subtraction have equal precedence, similarly division and multiplication have equal precedence) are calculated in left-to-right order. 

Example1: Evaluate 2 + 4 x 3

Solution: 2 + 4 x 3 = 2 + 12 = 14 (multiplication has higher precedence to addition)$\mathrm{.}$

Example2: Evaluate \(2 \times 4^3\)

Solution: \(2 \times 4^3 = 2 \times 64 = 128\) (exponent has higher precedence to multiplication).

Example 3: Evaluate \((2+4) \times 3\)

Solution: \((2+4) \times 3 = (6) \times 3 = 18\) (Parenthesis has higher precedence to multiplication).

Example 4: \(8 \div 2(2+2)=?\)

Solution: \(8 \div 2(2+2)=8 \div 2(4)=4 (4) = 16\)

Example 5: Evaluate \(10 \times 4-2 \times\left(4^2 \div 4\right) \div 2 \div 1 / 2+9\)

Solution:

\(
\begin{aligned}
&10 \times 4-2 \times\left(4^2 \div 4\right) \div 2 \div 1 / 2+9 \\
&=10 \times 4-2 \times(16 \div 4) \div 2 \div 1 / 2+9 \\
&=10 \times 4-2 \times(4) \div 2 \div 1 / 2+9 \\
&=40-8 \div 2 \div 1 / 2+9 \\ 
&=40-4 \div 1 / 2+9 \\
&=40-8+9 \\
&=40-8+9 \\
&=32+9 \\
&=41
\end{aligned}
\)

Example 6: Evaluate \(-10 \div\left(20 \div 2^2 \times 5 \div 5\right) \times 8-2\)

Solution:

\(
-10 \div\left(20 \div 2^2 \times 5 \div 5\right) \times 8-2
\)
First, stay inside the parenthesis until you get a single number
\(
\begin{aligned}
&=-10 \div(20 \div 4 \times 5 \div 5) \times 8-2 \\
&=-10 \div(5 \times 5 \div 5) \times 8-2 \quad(20 \div 4=5) \\
&=-10 \div(25 \div 5) \times 8-2 \quad(5 \times 5=25)
\end{aligned}
\)
Now again, resist the urge to do the multiplication first and remember that a negative divided by a positive is a negative.
\(
\begin{aligned}
&=-10 \div(5) \times 8-2 \\
&=-2 \times 8-2 \\
&=-16-2 \\
&=-18
\end{aligned}
\)

Example 7: \(9-3 \div \frac{1}{3}+1\)

Solution: \(9-9+1=0+1=1\)

Example 8: Evaluate \(4^2-6 \times 2 \div 4 \times 3+5\)

Solution: 

Do exponent:
\(
16-6 \times 2 \div 4 \times 3+5
\)
Multiply and divide from left to right
\(
\begin{aligned}
&=16-12 \div 4 \times 3+5 \\
&=16-3 \times 3+5
\end{aligned}
\)
\(
=16-9+5
\)
Add and subtract from left to right
\(
=16-9+5
\)
\(
=7+5=12
\)

Example 9: Evaluate \(\left(2+5^2\right)+4 \times 3-10\)

Solution:

Do parenthesis:
\(
(2+25)+4 \times 3-10
\)
\(
=27+4 \times 3-10
\)
Do multiplication
\(
=27+12-10
\)
Do Addition and then subtraction
\(
=39-10=29
\)

Example 10: Evaluate \(6 \times 4 \div 12+72 \div 8-9\)

Solution:

\(
\begin{aligned}
&6 \times 4 \div 12+72 \div 8-9 \\
&=24 \div 12+72 \div 8-9 \\
&=2+72 \div 8-9 \\
&=2+9-9 \\
&=11-9 \\
&=2
\end{aligned}
\)