# Simplyfying (1+(1/x)/(1+1/x))×(1+(1/x)/(1−1/x))

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• Yazan975
In summary, the expression (1+(1/x)/(1+1/x))×(1+(1/x)/(1−1/x)) can be simplified to 1+1/x, which represents any real number except x=0. It is important to simplify mathematical expressions for better understanding and quicker calculations.
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$\left(1+ \frac{\frac{1}{x}}{1+\frac{1}{x}}\right)\times\left(1+ \frac{\frac{1}{x}}{1- \frac{1}{x}}\right)$

Multiply both numerator and denominator of $\frac{\frac{1}{x}}{1+ \frac{1}{x}}$ by x to get $\frac{1}{x+ 1}$ and both numerator and denominator of $\frac{\frac{1}{x}}{1- \frac{1}{x}}$ to get $\frac{1}{x- 1}$. Now we have $\left(1+ \frac{1}{1+ x}\right)\times \left(1+ \frac{1}{x- 1}\right)$.Add the fractions in the left parentheses: $\left(1+ \frac{1}{1+ x}\right)= \left(\frac{1+ x}{1+ x}+ \frac{1}{1+ x}\right)= \left(\frac{x+ 2}{1+ x}\right)$

Add the fractions in the right parentheses: $\left(1+ \frac{1}{x- 1}\right)= \left(\frac{x- 1}{x- 1}+ \frac{1}{x- 1}\right)= \left(\frac{x}{x- 1}\right)$.So now the product is $\left(\frac{x+ 2}{1+ x}\right)\left(\frac{x}{x- 1}\right)= \frac{x^2+ 2x}{x^2- 1}$

## 1. What is the simplified form of the expression (1+(1/x)/(1+1/x))×(1+(1/x)/(1−1/x))?

The simplified form is 1+1/x.

## 2. How do you simplify the expression (1+(1/x)/(1+1/x))×(1+(1/x)/(1−1/x))?

To simplify, first simplify the expressions in the parentheses, then multiply the two resulting expressions together. This will result in 1+1/x.

## 3. Can the expression (1+(1/x)/(1+1/x))×(1+(1/x)/(1−1/x)) be further simplified?

No, the expression is already in its simplest form of 1+1/x.

## 4. What does the variable x represent in the expression (1+(1/x)/(1+1/x))×(1+(1/x)/(1−1/x))?

The variable x represents any real number, and the expression holds true for all values of x except x=0.

## 5. Why is it important to simplify mathematical expressions?

Simplifying mathematical expressions makes it easier to understand and work with them, and it also allows for quicker and more accurate calculations.

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