Download Project 4: Simplifying and Rewriting Expressions

MAT 51 Wladis Project 4: Simplifying and Rewriting Expressions Simplifying expressions What does it actually mean to simplify an expression? You may recall that we can rewrite an expression if and only if we replace that expression with an equivalent expression. So to find the simplest expression, we need to imagine all the possible expressions that might be equivalent to the expression that we have, and consider which of these would be the shortest version (with the fewest number of variables, operations and numbers) with the smallest numbers. Sometimes there is more than one possibility. In this class, we are only working with a small set of possible operations, so eventually once we get enough practice in with these types of operations, we will be able to see if problems can be simplified or not (at least after exploring a few possibilities). However, it is interesting to note that in math in general, trying to find the simplest form of something is actually a very complex (and in many cases even unsolvable) problem. There are some cases that you might encounter later in different math classes (if you continue to take math for several years) where you will see that it is not possible to determine if something is completely simplified or not. Examples: A) Consider the following expression: (where 0) This expression could be rewritten in an infinite number of ways, but here are just a few (we will talk about HOW to do this rewriting in the next section—for now you can simply take our word for it. ): i.
This is a simplified version of the expression, because: none of the numbers can be simplified or
reduced further, none of the variables even occur more than once, so they definitely can’t be reduced, and
there is no way for us to get rid of the fraction bar without making the expression even more complicated,
except for the option ii below, which is also simplified. This has the same numbers and variables as i. above. This expression doesn’t have a fraction bar
ii. 2
like i., but it does have a negative sign that i. doesn’t have, so in the end, we have the same number of
numbers/variables/operations here as we do in i. So both i. and ii. are reduced versions of the original
expression. iii.
This version is more simplified than the original version (the x on top no longer has an exponent),
but it is not as simplified as i. and ii. above, because we have two instances of the x here—on in the
numerator and one in the denominator, whereas in i. and ii., we only have one instance of x in the
expression. iv.
This is also not simplified, because the numbers here are larger than in i. or ii, since we have an 8 on
top and a 4 on the bottom as coefficients at the left of the expression. (Also, we have one number more
than in i. or ii.). 1. Consider the following expression: 2
3 3
1 All of the following expressions are equivalent to this expression—which of these is the most simplified? There may be more than one. Circle the one(s) that is(are) the most simplified, and briefly explain why. i.
2
3 ⋅3
2
3 ii.
2 ⋅3
2
iii.
6
2
iv.
6
11
3
3⋅3
3
9
3
,
2. Consider the following expression: 0 All of the following expressions are equivalent to this expression—which of these is the most simplified? There may be more than one. Circle the one(s) that is(are) the most simplified, and briefly explain why. ,
0 i.
2
ii.
,
0 iii.
,
0 iv.
0 ,
3. Consider the following expression: 2
⋅
,
0 All of the following expressions are equivalent to this expression—which of these is the most simplified? There may be more than one. Circle the one(s) that is(are) the most simplified, and briefly explain why. i.
2
,
0 ,
0 ii.
2
iii.
√2
0 ,
iv.
2
,
0 4. Consider the following expression: 5√5 √5 2√5 All of the following expressions are equivalent to this expression—which of these is the most simplified? There may be more than one. Circle the one(s) that is(are) the most simplified, and briefly explain why. i.
6√5 ii.
4√5 2√5
iii.
4
iv.
7√5
2 √5
√5
Simplifying expressions, or rewriting them with a specific goal in mind Typically when we rewrite expressions, it is with a specific goal in mind. We may want to simplify them, or we may want to put them into a particular form that allows us to see relationships (for example, when we come to equations, we often put them in a particular form in order to be able to solve them or graph them). When we have a goal in mind for rewriting an expression (or equation), we then have to put together many of the things that we have already been doing so that we apply identities to the expressions one after another. This requires several steps: 1) We have to identify our end goal (e.g. Why are we rewriting the expression/equation? What does it need to look like when we are done?). 2) We have to select particular identities that could be applied to the current expression (either in its current form, or with a little rewriting), and we need to consider each of these possible identities as a way of rewriting the expression. If the rewriting is somewhat complex, this may require us to set some smaller goals that we need to meet along the way to meet our larger goal. 3) Each time we apply an identity to the expression, we have to ask ourselves if this has brought us closer to our goal or not. a. If not, we may need to put that work aside and to start over by applying a different identity to the original equation. b. If it has gotten us closer, then we need to think about what identity we should apply to this new rewritten function in order to get even closer to our goal. 4) We keep repeating this process until we reach our goal. As problems grow more complex, we might have to apply many, many identities sequentially to get to our final result. Let’s just jump in and start trying to simplify expressions by using the identities we have been using so far. To keep things simple, we will only need to use the identities on the list below as we work to simplify the next set of problems. 










1 ⋅ …⋅

1

⋅ 
0 ⋅ 
0 ,

√
√ √ 
√
0 0 0 Examples: B) Rewrite the following expression, with the aim of simplifying it as much as possible: 6
5
2
6 If our goal is to simplify this, we will likely want to get rid of the parentheses so that we can try to combine what
is in each set of parentheses with each other. If that is possible, it will very likely make the expression simpler. So,
our first mini-goal is: Rewrite the expression without parentheses.
If the expression was of this form:
, then we could use the identity
to rewrite
this expression without the first set of parentheses. But we need to rewrite all of the subtraction in this expression
as addition first. So this leads us to another mini-goal, which must come BEFORE our goal of rewriting the
expression without parentheses: Rewrite all subtraction in the expression as addition.
Let’s put these two goals in the order that we need to do them, and list the identities needed to accomplish each
goal:
Mini-goal
Identities needed
1) Rewrite all subtraction in the expression as addition.
2) Rewrite the expression without parentheses. We will
start with the first set of parentheses and then focus on the
second set.
Now let’s try to do these two steps, and see what happens:
Step 1: Rewrite the subtraction in the expression as addition, using the identity
6
5 ,
2
6 → → → 6
5
2
6
6
5
2
6
6
5
So 6
5
2
2
6 6
Step 2: Rewrite the expression without (the first set of) parentheses, using the identity
Focusing on rewriting without the first set of parentheses:
6 ,
5,
2
6 →
→ → 6
5
So 6
6
6
2
5
2
6
6
5
.
5
2
2
6
6
Now we need to focus on rewriting the expression without the second set of parentheses. The sub-expression
2
6 almost looks like the left side of the identity
, except that the variable can’t
stand in for a negative sign alone—the negative sign has to belong to a number or an expression. So now we form
some new mini-goals, on our way to the goal of rewriting the expression without parentheses:
into the form
.
Mini-goal: Put
We can do this if we notice that we can apply the identity
1 to the expression 2
First let’s rewrite the identity
1 with another variable to avoid confusion:
1 .
6 , → 2
So 6
5
2
1 → 6
6
1 → 2
5
1 2
6
6
1 2
6
6 . So, let’s do that:
Now, back to our first mini goal:
Mini-goal: Rewrite the expression without the second set of parentheses.
We see that we can now use the identity
to rewrite the second half of the expression without
parentheses:
1,
→ 2
1
,
2
→ 1 2
6
5
So 6
6 → 1 2
6
1 6
6
5
1 2
1 6
So now we have simplified the original expression to this equivalent expression: 6
5
1 2
1 6
How can we simplify this expression further? The most obvious step is for us to perform the multiplication in the
last few terms:
6
5
1 2
1 6
6
5
1 2
6
6
6
5
5
6 because
1 2
2
6 because
1 6
6
1⋅2⋅
1 2
2
So now we have simplified the original expression to this equivalent expression: 6
5
2
6
What might help to make this simpler? It would be good for us to look to see if we can combine any of these terms
together. For example, if 5 and 6 were next to each other, we could add them. Similarly, if 6 and 2 were
next to each other, we could use the identity
to combine them. So, our next mini-goal is:
Mini-goal: Reorder the terms so that “like” terms are next to each other.
I can use the identity
to do this:
5,
2
So 6
5
→ → 2
6
6
2
5
→ 5
2
2
5
6
Now we can simply add the final two like terms:
2
5
6
6
2
1 because 5
6
6
1
And now our final mini-goal is:
Mini-goal: Used the identity
to combine the first two terms.
We noted that the first two terms have the form of the right side of the identity.
6,
→ 6
2,
→ 2
6
2
So, 6
1
6
→ 1
2
4
2
2
And finally, we can add the 6 and the
6
1 because 6
2
6
2
2 inside the parentheses, so we go ahead and do that:
1 because 6
2
4
1
So now we have replaced the original expression with the equivalent expression 4
This is about as simple as it gets, although, we could do one final step and rewrite the addition of a negative as
subtraction, if we want:
4
So, 4
,
1
1 → 4
1.
→ → 4
1
This is definitely as simple as we could make this expression, so our final expression is 4
6
5
2
6
4
1 4
1 and:
1
5. Rewrite the following expression, with the aim of simplifying it as much as possible: 2
3
3
1 6. Rewrite the following expression, with the aim of simplifying it as much as possible: 2
3 3
1