Welcome to a lesson on order of operations.
If we are working with a mathematical expression
that contains more than one operation,
we need to understand how to simplify,
and the order of operations give us the rules to follow.
The acronym PEMDAS is meant to help us remember
the steps in the order of operations.
But we do need to be careful here,
because notice how there are six letters in PEMDAS
and only four steps in the order of operations.
Where step one, represented by P,
tells us simplify inside the parentheses or brackets
or any grouping symbols first.
Step two, represented by E,
tells us to simplify the exponents and roots next.
Notice for step three we have two letters.
We have MD, which tells us to simplify
the multiplication and division as it occurs
from left to right.
So we don't multiply first and then divide.
These two operations have the same priority,
so you multiply and divide from left to right.
And then finally step four we have the letters AS,
which tell us to simplify the addition and subtraction
from left to right.
Again, addition and subtraction have the same priority.
Let's look at some examples.
Notice how these first two expressions look almost the same,
but in this expression here we have parentheses
around two times five.
And since the first step in the order of operations
is to simplify inside the parentheses or brackets,
we want to find this product first.
Two times five equals 10,
so this simplifies to 10 squared.
10 squared equals 10 times 10, which equals 100.
So this first expression simplifies to 100.
Now for the next expression notice how there no parentheses,
so we got to step two where we simplify the exponents.
So we're going to simplify five squared first.
Five squared equals five times five, which equals 25,
so this simplifies to two times 25, which is equal to 50.
So this expression simplifies to 50.
Let's go and check these first two on the calculator.
First we have in parentheses two times five,
and this is squared, which does equal 100.
For the second expression we have two times five squared,
which does equal 50.
Now again the next two expressions look almost the same,
but the second expression here has parentheses
around the sum of seven and one.
But for this expression here notice how we have
subtraction and addition, which have the same priority,
so we had and subtract from left to right.
So we find this difference first,
10 minus seven equals three.
So this simplifies to three plus one,
which is equal to four.
But for the next expression we simplify
inside the parentheses first,
so we'll first find the sum seven plus one equals eight.
So this simplifies to 10 minus eight, which is equal to two.
And again checking on the calculator,
we have 10 minus seven plus one for this first expression,
which does equal four.
And for this expression here we have 10
minus the quantity seven plus one,
which does equal two.
Now let's look at three more examples.
Looking at example two, the first step is to simplify
inside the parentheses, so we find this difference first.
So we have 24 divided by,
four minus two is two, so we have divided by two cubed.
Next step we simplify the exponents,
so we simplify two to the third.
Two to the third equals two times two times two,
which equals eight.
So now we have 24 divided by eight,
and 24 divided by eight equals three.
Checking on the calculator,
we have 24 divided by the quantity four minus two,
close parenthesis, and this is raised to the third power
so here's the exponent key or the caret, three, enter,
verifying our answer.
In example three we simplify inside the parentheses first,
but notice inside the parentheses we have
addition and division,
and notice how division has higher priority than addition,
so we find this quotient first.
So we have four plus five times the quantity one
and then plus 12 divided by six is equal to two,
and this is squared.
We're still simplifying inside the parentheses,
so now we find this sum.
So we have four plus five times one plus two is three,
so we have times three squared.
Now we simplify the exponents,
three squared is equal to nine.
So we have four plus five times nine.
Now we have addition and multiplication,
and we multiply before we add,
so the next step is to find this product here.
Five times nine is equal to 45,
so we have four plus 45, which is equal to 49.
Back to the calculator,
we have four plus five times the quantity
one plus 12 divided by six, closed parenthesis, squared,
and enter, verifying 49 is correct.
And now for our last example,
notice how we have an expression in fraction form.
Whenever simplifying a fraction there are
implied parentheses around the numerator and denominator.
So when we have an expression in fraction form,
we want to simplify the numerator and denominator separately
and then simplify the fraction, or perform the division.
So 15 minus three is equal to 12,
and one plus five is equal to six,
and 12 divided by six is equal to two.
Now checking on the calculator, it is important to include
parentheses around the numerator and denominator.
So open parenthesis, 15 minus three, close parenthesis,
divided by, open parenthesis,
one plus five, close parenthesis, and enter,
verifying our answer.
I hope you found this helpful.