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
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. $\frac{x}{1 - \left( \frac{x - 1}{x} \right)}$
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 $a(b + c) = ab + ac$. It is used to simplify expressions and to remove parentheses.
Multiplying Fractions: The multiplication of fractions is straightforward: $\frac{a}{b} \cdot \frac{c}{d} = \frac{ac}{bd}$. 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 $x^2$.
