Welcome to a lesson
on evaluating algebraic expressions.
To evaluate an algebraic expression,
sometimes called a variable expression,
the first step is to substitute
or replace the given value
or values of the variable or variables
into the expression.
In step two, we evaluate using the order of operations.
For review, the order of operations are listed below.
For step one, we simplify inside
the parentheses or grouping symbols.
Step two, we simplify the exponents and roots.
Step three, we multiply and divide from left to right.
And step four, we add and subtract from left to right.
Let's look at some examples.
We want to evaluate all the given expressions
when w equals two,
which means for each expression,
we're going to substitute or replace w with two,
here,
here, and so on.
So the expression w minus six is equal to
two minus six when w equals two
and two minus six equals negative four.
The expression six minus w is equal to
six minus two when w equals two
and six minus two is equal to four.
The expression five w minus three is equal to
five times two minus three when w equals two.
Here we have multiplication and subtraction.
We multiply before subtracting,
so five times two is equal to 10,
so this simplifies to 10 minus three
and 10 minus three simplifies to seven.
Here we have w to the third power,
which is equal to two to the third power
when w equals two
because the exponent is three,
we have three factors of two,
so this is equal to two times two times two,
which is equal to eight.
Next we have three w to the second
or three w-squared,
so when w equals two
this is equal to three times two-squared.
Next we simplify the exponents
two squared or two to the second
is equal to two times two,
which equals four.
So, this simplifies to three times four
and three times four is equal to 12.
So on top of this example,
the exponent of two is only attached to the w,
but our next expression,
because three w is in parentheses,
this means three w times three w.
So when w equals two,
we have, in parentheses, three times two
and this is raised to the second power.
Simplifying inside the parentheses first,
three times two is equal to six,
so this simplifies to six squared
and six squared or six to the second
is equal to six times six,
which equals 36.
Next we have four over five w
and because a fraction bar means division
we can think of this as four divided by five w,
but when w equals two,
this would be four divided by five times two.
Now when simplifying an expression
that contains a fraction,
we simplify as if there are a set of parentheses
around a numerator and denominator.
So, we simplify the numerator and denominators first
and then we simplify the fraction
or perform the division indicated by the fraction bar.
So, notice how the numerator
of four does not simplify.
Our denominator is five times two,
which equals 10,
so this expression is equal to 4/10
when w equals two,
but this fraction does simplify
because four and 10 share a common factor of two.
If we divide out the common factor of two,
the expression simplifies to 2/5.
Another way to show the simplification of 4/10
would be to write four as two times two
and write 10 as five times two,
so we can actually see the common factor of two.
Two divided by two simplifies to one
giving us a simplified fraction of 2/5.
Next, we have five w over four
or five w divided by four,
so when w equals two,
we have five times two over four.
Simplifying the numerator,
five times two is equal to 10,
so we have 10/4.
Again, we have a common factor of two,
which we can divide out
giving us a simplified fraction of 5/2
or again, we can also show the simplification
by writing 10 as five times two
and four as two times two.
We can actually see the common factor of two.
Two divided by two simplifies to one
leaving us with the simplified fraction of 5/2.
And here we have three raised to the power of w,
so when w equals two,
we have three to the second power
or three squared,
which is equal to three times three,
which equals nine.
Now to go to example two.
We want to evaluate a b plus c
given a equals negative five,
b equals seven,
and c equals negative three.
So, we'll substitute negative five for a,
seven for b,
and negative three for c.
Remember a b means a times b,
so performing substitution,
we'd have the expression negative five times seven
plus negative three.
We have multiplication and addition.
We multiply first.
Negative five times seven
is equal to negative 35.
So, we have negative 35 plus negative three
what is equal to negative 38.
So, the given expression equals negative 38
given the value of the variables.
Clicking on the calculator,
we have open parenthesis negative five times seven
plus
negative three,
which verifies our work.
Next we have the quantity a squared minus b squared
when a equals negative five
and b equals negative three.
So, we substitute negative five for a
and negative three for b.
So, we'd have in parentheses
negative five squared
minus, in parentheses,
negative three squared.
And it is important to have parentheses
around these negative bases,
which we'll talk about in a moment.
Next, we want to simplify the exponents,
so negative five squared means
negative five times negative five,
which equals positive 25.
And we have minus negative three-squared,
which equals negative three times negative three,
which equals nine
and 25 minus nine is equal to 16.
Before we look at our last example though,
it is important to recognize
that if we have negative five in parentheses squared
and compare this to negative five squared,
these are different.
Here the base is negative five,
so we have two factors of negative five,
which gives us positive 25.
And here we have the opposite
of positive five-squared,
so the base here is positive five,
so this is negative positive five
times positive five,
which equals negative 25.
So, it is important we recognize difference
between these two expressions
and it's also true on the calculator.
Notice how if we have in parentheses
negative five-squared,
this gives us positive 25,
but if we have negative five squared,
which means the opposite of positive five-squared,
we do get negative 25.
Now for our last example,
a local window washing company charges $11.92
for each window plus a reservation fee of $7.
We're asked to write an algebraic expression
to represent the total cost
for the window washing company
for washing w windows.
Because it cost $11.92 for each window,
plus the reservation fee of $7,
assuming the expression is in dollars,
the expression for the total cost
would be
11 point nine two
times w,
so we have the cost per window times w,
which represents the number of windows
plus a $7 reservation fee.
For part b,
we're asked to use the expression here
to determine the total cost
of washing 17 windows.
So, we substitute 17 for w,
which would give us 11 point nine two
times 17
plus seven.
Evaluating this expression,
we multiply first,
so we'd have 11 point nine two
times 17,
which gives us 202 point six four.
You'll still have plus seven.
202 point six four plus seven
is equal to 209 point six four,
which means the total cost
for washing 17 windows
is 209 dollars and 64 cents.
I hope you found this helpful.