Intuition about simplifying this expression

In summary, the conversation discusses a problem where an expression can be simplified to -e-t, but the person was unable to recognize this and continued using a more complicated method. They wonder if this is due to inexperience and ask for advice on becoming a stronger mathematician. Another person suggests changing the symbols to make the simplification more apparent.
  • #1
Ikari
5
0

Homework Statement


I was solving a problem and came to a part where I was dealing with the expression,

1 - e-t
--------
1 - et

It turns out that this can be simplified to -e-t but I had no idea from simply looking at the first expression that it could be simplified, and I tried to continue using a cludgy quotient rule to get the derivative, where if I had have simplified it, it would have been much easier. Is this simply a matter of being inexperienced, or am I missing some concept in elementary algebra that would let me recognize a potential simplification? For instance, I can easily recognize difference of squares, etc, but not something like this. How can I become a stronger mathematician moving forward in this regard?

Thanks in advance!

Homework Equations


N/A

The Attempt at a Solution


N/A[/B]
 
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  • #2
Ikari said:

Homework Statement


I was solving a problem and came to a part where I was dealing with the expression,

1 - e-t
--------
1 - et

It turns out that this can be simplified to -e-t but I had no idea from simply looking at the first expression that it could be simplified, and I tried to continue using a cludgy quotient rule to get the derivative, where if I had have simplified it, it would have been much easier. Is this simply a matter of being inexperienced, or am I missing some concept in elementary algebra that would let me recognize a potential simplification? For instance, I can easily recognize difference of squares, etc, but not something like this. How can I become a stronger mathematician moving forward in this regard?

Thanks in advance!

Homework Equations


N/A

The Attempt at a Solution


N/A[/B]
Don't beat yourself up about this. We all have cases where the answer seems obvious after we see what it is.
 
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Likes Ikari
  • #3
Ikari said:

Homework Statement


I was solving a problem and came to a part where I was dealing with the expression,

1 - e-t
--------
1 - et

It turns out that this can be simplified to -e-t but I had no idea from simply looking at the first expression that it could be simplified, and I tried to continue using a cludgy quotient rule to get the derivative, where if I had have simplified it, it would have been much easier. Is this simply a matter of being inexperienced, or am I missing some concept in elementary algebra that would let me recognize a potential simplification? For instance, I can easily recognize difference of squares, etc, but not something like this. How can I become a stronger mathematician moving forward in this regard?

Thanks in advance!

Homework Equations


N/A

The Attempt at a Solution


N/A[/B]

Sometimes it helps to change the symbols: let ##y = e^t##. Then your fraction is
$$ \text{fraction} = \frac{1 - \frac 1 y}{1-y} = \frac{ \frac{y-1}{y}}{1-y}.$$
Can you see now how the simplification would proceed?

Anyway, after simplifying, put back ##e^t## in place of ##y,## (or ##e^{-t}## in place of ##1/y##).
 
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Related to Intuition about simplifying this expression

1. What is the purpose of simplifying an expression?

The purpose of simplifying an expression is to make it easier to work with and understand. It involves reducing the expression to its simplest form by combining like terms, canceling out common factors, and applying other algebraic rules.

2. How do I know when an expression can be simplified?

An expression can be simplified when it contains terms that can be combined or factors that can be canceled out. It is also helpful to look for patterns or common factors in the expression.

3. What are the common techniques for simplifying expressions?

Some common techniques for simplifying expressions include combining like terms, using the distributive property, factoring, and canceling out common factors. These techniques can help to reduce the number of terms and variables in an expression.

4. What are some tips for simplifying expressions more efficiently?

One tip for simplifying expressions more efficiently is to work from the inside out, starting with the innermost parentheses or brackets and simplifying those terms first. It is also helpful to use mental math and to look for shortcuts, such as identifying common factors or using the distributive property.

5. How can simplifying expressions be useful in real-world applications?

Simplifying expressions is useful in real-world applications, such as in scientific and mathematical calculations, budgeting, and problem-solving. It can help to make complex problems more manageable and provide a clearer understanding of the underlying relationships between variables and constants.

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