Now, we'll look at two examples
of simplifying expressions involving integers.
To simplify an expression,
we follow the order of operations, outlined here.
Notice there are four steps in the order of operations.
Step one, we simplify within the innermost parentheses,
or grouping symbols.
Looking at our expression,
we have two sets of grouping symbols.
We have the outer brackets, and the inner parentheses,
so because we're supposed to simplify
within the innermost parentheses or grouping symbols first,
the first step is to simplify within these parentheses.
Notice, within the parentheses, we have an exponent of two,
as well as subtraction.
So, referring back to the order of operations,
notice how we simplify exponents
before we add and subtract, so for our first step,
we want the simplify the square of negative three.
So the given expression is equal to two, minus five,
times the quantity, one plus six, times the quantity,
the square of negative three means
we have two factors of negative three,
which equals negative three times negative three,
which equals positive nine.
So, within the parentheses, we have nine minus four.
We're still simplifying within the parentheses.
The next step is to perform this subtraction.
So, we have two minus five, times the quantity,
one plus six, times, nine minus four equals five.
Now, we've simplified within the innermost parentheses,
but we still have these brackets,
which are also grouping symbols,
so we're still on step one of the order of operations.
Now, we simplify within the brackets.
Notice, within the brackets, we have multiplication,
as well as addition.
So, while it might be tempting to find this sum next,
that would not be correct,
because looking at the order of operations,
notice how we multiply and divide
before we add and subtract.
So, within the brackets,
our next step is to find this product.
So, we have two minus five, times the quantity,
one plus, six times five is 30.
Still simplifying inside the brackets, we now find this sum.
So, we have two minus five, times 31.
So finally, we're down to two operations.
We have subtraction, as well as multiplication.
And once again, notice how we multiply and divide
from left to right before we add and subtract.
So, the next step is to find this product.
So, we'd have two minus, five times 31 is equal to 155,
and then, finally, we have one operation left.
We have subtraction,
which notice how is step four of the order of operations,
add and subtract from left to right.
So, two minus 155 equals negative 153.
I should mention, if it's helpful,
we could rewrite two minus 155,
as two plus negative 155,
but of course, the result is still negative 153.
Let's look at a second expression
that's similar to this expression.
Again, the directions are the same.
We want to evaluate the given expression.
So step one, simplify within the innermost
parentheses or grouping symbols.
So, we simplify within these parentheses here,
so in the parentheses, again, we have an exponent of two,
and addition, and therefore, within the parentheses,
we simplify the exponent first.
So, we have six plus two, times the quantity,
negative three, minus four, times,
the square of negative one equals
negative one times negative one, which is positive one.
So, within the parentheses, we have one plus eight.
Still simplifying inside the parentheses,
notice that one plus eight is nine,
so we have six plus two, times the quantity,
negative three, minus four, times nine.
We're still on step one, but now we simplify
within the brackets, which are grouping symbols.
Notice, within the brackets,
we have subtraction and multiplication.
Again, just to review, notice how we multiply and divide
before we add and subtract,
so the next step is to find this product.
So, we have six plus two, times the quantity,
negative three, minus, four times nine is equal to 36,
so inside the brackets, we have negative three minus 36.
Still simplifying inside the brackets,
negative three minus 36 equals negative 39.
So, we have six plus two times negative 39.
We have addition and multiplication, and just as before,
we multiply and divide before we add and subtract.
So, the next step is to find this product.
A positive times a negative is negative, and therefore,
two times negative 39 equals negative 78.
And then, finally, our last step, we find this sum.
Negative six plus negative 78, which if we wanted to,
we could write as six minus 78.
Either way, we get negative 72.
So, these are two very good examples of verifying
that we have a good understanding
of the order of operations,
as well as operations involving integers.
I hope you found this helpful.