Simplify x/(1-1/(1+1/(x-1)))

The question asks for the simplification of a mathematical expression that consists of a fraction and several nested fractional parts, specifically in the form of a continued fraction. The algebra involved would typically require the application of fraction operations such as finding a common denominator, simplification by performing addition or subtraction of fractions, and possibly rearranging the expression to isolate the variable x in order to simplify the entire expression into a more manageable and reduced form.

$\frac{x}{1 - \frac{1}{1 + \frac{1}{x - 1}}}$

Answer

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Solution:

Step:1

Transform the denominator into a simpler form.

Step:1.1

Express $1$ as a fraction with a common denominator. $\frac{x}{1 - \frac{1}{\frac{x - 1}{x - 1} + \frac{1}{x - 1}}}$

Step:1.2

Combine the terms in the numerator over the shared denominator. $\frac{x}{1 - \frac{1}{\frac{x}{x - 1}}}$

Step:1.3

Simplify the fraction inside the denominator.

Step:1.3.1

Combine like terms. $\frac{x}{1 - \frac{1}{\frac{x}{x - 1}}}$

Step:1.3.2

Express $x$ as a sum with zero. $\frac{x}{1 - \frac{1}{\frac{x}{x - 1}}}$

Step:2

Refine the denominator further.

Step:2.1

Multiply the numerator by the inverse of the denominator. $\frac{x}{1 - (\frac{x - 1}{x})}$

Step:2.2

Multiply $\frac{x - 1}{x}$ by $1$ . $\frac{x}{1 - \frac{x - 1}{x}}$

Step:2.3

Convert $1$ to a fraction with the same denominator. $\frac{x}{\frac{x}{x} - \frac{x - 1}{x}}$

Step:2.4

Add the numerators over the common denominator. $\frac{x}{\frac{1}{x}}$

Step:2.5

Express the fraction in the denominator in a simplified form.

Step:2.5.1

Apply the distributive property. $\frac{x}{\frac{1}{x}}$

Step:2.5.2

Simplify the expression. $\frac{x}{\frac{1}{x}}$

Step:2.5.3

Reduce the terms. $\frac{x}{\frac{1}{x}}$

Step:2.5.4

Finalize the simplification. $\frac{x}{\frac{1}{x}}$

Step:3

Multiply the numerator by the reciprocal of the denominator. $x \cdot x$

Step:4

Compute the product of $x$ with itself. $x^{2}$

Knowledge Notes:

To solve the given problem, we need to simplify a complex rational expression. The key steps in this process include:

Common Denominator: Finding a common denominator is essential when dealing with complex fractions. It allows us to combine fractions into a single term.
Simplifying Complex Fractions: This involves rewriting nested fractions into a simpler form by combining terms and using the property that $\frac{1}{\frac{a}{b}} = \frac{b}{a}$ .
Distributive Property: The distributive property states that $a (b + c) = a b + a c$ . It is used to simplify expressions and to remove parentheses.
Multiplying Fractions: The multiplication of fractions is straightforward: $\frac{a}{b} \cdot \frac{c}{d} = \frac{a c}{b d}$ . When multiplying a fraction by its reciprocal, the result is 1.
Simplifying Algebraic Expressions: This involves combining like terms and reducing expressions to their simplest form.
Multiplication of Variables: When multiplying a variable by itself, we use exponents to express the product. For example, $x \cdot x = x^{2}$ .

By following these steps and applying these principles, we can simplify the given complex rational expression to $x^{2}$ .