Discovering Cancellations: $(1+x)^2/(1-x^2)$

[soroban]

You have probably seen these cancellations.

$\displaystyle\quad \frac{16}{64} \;=\;\frac{1\rlap{/}6}{\rlap{/}64} \;=\;\frac{1}{4}$

$\displaystyle\quad \frac{26}{65} \;=\;\frac{2\rlap{/}6}{\rlap{/}65} \;=\;\frac{2}{5}$

$\displaystyle\quad \frac{19}{95} \;=\;\frac{1\rlap{/}9}{\rlap{/}95} \;=\;\frac{1}{5}$But have you seen this one?

$\displaystyle \quad \frac{(1+x)^2}{1-x^2} \;=\;\frac{(1+x)^{\rlap{/}2}}{1-x^{\rlap{/}2}} \;=\;\frac{1+x}{1-x}$

 

[lofgran]

Dear forum members,

Thank you for bringing these cancellations to my attention. I am always fascinated by patterns and relationships in mathematical equations. These cancellations are a great example of how seemingly complex expressions can be simplified and reduced to more manageable forms.

The first three cancellations you shared are known as "hidden fractions" or "hidden cancellations." They occur when the numerator and denominator of a fraction share a common factor that can be reduced. In the first example, both the numerator and denominator can be divided by 16, resulting in a simplified fraction of 1/4.

Similarly, in the second and third examples, the numerators and denominators can be divided by 13 and 19, respectively, resulting in simplified fractions of 2/5 and 1/5.

However, the last cancellation you shared is a bit different. It involves the use of fractional exponents, which can also lead to cancellations. In this case, both the numerator and denominator can be rewritten as fractional exponents with the same base, allowing for the cancellation to occur.

These types of cancellations may seem surprising, but they are actually based on fundamental principles of algebra and can be proven mathematically. They are also commonly used in simplifying and solving equations in higher level mathematics.

Thank you again for bringing these cancellations to my attention. I hope this explanation has helped shed some light on the topic and sparked your interest in exploring more mathematical patterns and relationships.