\(\displaystyle{ x + 3 / 2 = x + \frac{3}{2} }\)
If you want the entire x + 3 expression to be divided by 2, you need to use parentheses to override rule 3 taking precedence to rule 4.
\(\displaystyle{ (x + 3) / 2 = \frac{x+3}{2} }\)

e^2x = \( e^2x \)
If you want 2x to be the exponent to e, you need to use parentheses to override rule 2 taking precedence to rule 3. e^(2x) = \( e^{2x} \)

Here is one I see a lot. A student enters -3^2 into their calculator. This evaluates to -9 because the 2 power takes precedence to the negative sign. So this is like entering -(3^2). The negative sign falls in the same category as subtraction. We just usually do not write 0-3^2. To get (-3)^2, the parentheses are required.

Note: Take special care not to use computer notation in your written work. Good teachers require correct notation and computer notation is not considered correct notation for written work.

\( 3 \div 4 \times 16 \)
I think of the operation before a number (or variable) as being stuck to the number. Then I convert the division into multiplication. So to divide by 4 is the same multiplying by 1/4. The equation is now \(\displaystyle{ 3 \times \frac{1}{4} \times 16 }\). Now I just multiply everywhere and I can do it in any order. It makes more sense to me to multiply the last two since I will get an integer. So I have \( 3 \times 4 = 12 \). I can do this only because I replaced the division by multiplication. This technique will come in handy in calculus.

Now let's look at addition and subtraction. Again, I think of the operation as being stuck to the number it follows. So in \( 3 - 4 + 16 \) the subtraction sign in front of the 4 is stuck to the 4. I can move the subtraction and replace it with addition if I stick a negative sign to the 4.
\( 3 + (-4) + 16\) (The parentheses are required. 3 + - 4 does not make sense.) Now I have addition everywhere and I can do the additions in any order.
\( -1+16 = 3+12 = 15 \)

Unless otherwise instructed, simplify these expressions using the correct order of operations.