Order of Operations-3

There are several types of grouping symbols that can be used besides parenthesis. One type is a fraction bar. If we have a fraction, the entire numerator and the entire denominator must be evaluated before we reduce the fraction. In these cases we can simplify in both the numerator and denominator at the same time. Example `(2^4-(-8)times3)/(15divide5-1)`
`(color(red)(2^4)-(-8)times3)/(color(red)(15divide5)-1)` Exponent in the numerator, divide in denominator
`=(16-color(red)((-8)times3))/color(red)(3-1)` Multiply in the numerator, subtract in denominator
`=color(red)(16-(-24))/2` Add the opposite to simplify numerator, denominator is done.
`=40/2` Reduce, divide
`=20` Solution
Another type of value. When we have absolute value we will evaluate everything inside the absolute value, just as if it were a normal parenthesis. Then once the inside is completed we will take the absolute value, or distance from zero, to make the number positive. Example `1+3|−4^2 −(−8)|+2|3+(−5)^2|`
`1+3|−color(red)(4^2) −(−8)|+2|3+color(red)((−5)^2)|` Evaluate absolute values first, exponents
`=1+3|color(red)(−16−(−8))|+2|color(red)(3+25)|` Add inside absolute values
`=1+3color(red)(|−8|)+2color(red)(|28|)` Evaluate absolute values
`=1+24+color(red)(2(28))` Finish multiplying
`=color(red)(1+24)+56` Add left to right
`=25+56` Add
`=81` Solution
The above example also illustrates an important point about exponents. Exponents only are considered to be on the number they are attached to. This means when we see `− 4^2`, only the `4` is squared, giving us `− (4^2)` or `− 16`. But when the negative is in parentheses, such as `( − 5)^2` the negative is part of the number and is also squared giving us a positive solution, `25`.