Problem 81 - Algebra and Trigonometry

Question

Simplify the fractional expression. (Expressions like these arise in calculus.) $$ \sqrt{1+\left(\frac{x}{\sqrt{1-x^{2}}}\right)^{2}} $$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{1}{\sqrt{1-x^2}} \).

Step 1: Understand the Expression

We need to simplify the given expression: \[ \sqrt{1+\left(\frac{x}{\sqrt{1-x^{2}}}\right)^{2}} \].This is a combination of a square root and a fraction inside a complex radical expression.

Step 2: Simplify the Fraction

Let's simplify the fraction inside the square:\[ \frac{x}{\sqrt{1-x^{2}}} \].The square of this fraction is:\[ \left(\frac{x}{\sqrt{1-x^{2}}}\right)^2 = \frac{x^2}{1-x^2}. \]

Step 3: Substitute the Simplified Fraction

Now, substitute the squared fraction back into the original expression:\[ \sqrt{1 + \frac{x^2}{1-x^2}}. \]

Step 4: Simplify Inside the Square Root

Combine the terms inside the square root into a single fraction:\[ \sqrt{\frac{1(1-x^2) + x^2}{1-x^2}} = \sqrt{\frac{1-x^2 + x^2}{1-x^2}}. \]Notice that \( -x^2 + x^2 = 0 \), so it simplifies to:\[ \sqrt{\frac{1}{1-x^2}}. \]

Step 5: Simplify the Expression Further

Finally, take the square root of the fraction:\[ \sqrt{\frac{1}{1-x^2}} = \frac{1}{\sqrt{1-x^2}}. \]

Key Concepts

Fractional Expression

A fractional expression is a fraction in which the numerator and/or the denominator are algebraic expressions. In this exercise, the fractional expression is given as \( \frac{x}{\sqrt{1-x^2}} \). Understanding and simplifying such expressions is crucial because they frequently appear in more complex mathematical problems, especially in calculus.

Fractional expressions may appear in the context of functions or equations, and simplifying them often involves ensuring that the expression is in its simplest possible form. This can make it easier to perform other operations or interpret the meaning of the expression in a given mathematical context.

• **Numerator and Denominator**: The terms on the top and bottom of the fraction respectively.
• **Algebraic Expressions**: Can include variables, constants, exponents, and operations such as addition, subtraction, multiplication, and division.

By tackling the fractional part of \( \frac{x}{\sqrt{1-x^2}} \) directly within the larger expression, we could simplify it into a form where each part of the fraction holds clear definitions and roles in the complete expression. Understanding this fraction was a key step in our overall simplification.

Simplification Techniques

Simplification techniques are methods used to reduce a complex expression to its simplest form without changing its value. In the provided exercise, we aim to simplify a complex radical expression, which is a fraction under a square root sign. This involves several steps of algebraic manipulation:

• **Squaring fractions**: The first step involved squaring the fraction \( \left(\frac{x}{\sqrt{1-x^2}}\right)^2 \). This results in a simpler format \( \frac{x^2}{1-x^2} \), which is easier to work with.
• **Combining terms**: Inside the square root, the terms were combined into a single fraction \( \sqrt{\frac{1 - x^2 + x^2}{1-x^2}} \).
• **Cancelation of terms**: Notice here that \(-x^2 + x^2 \) cancels out to zero, simplifying the expression further.
• **Reducing complex fractions**: After cancellation, the expression became \( \sqrt{\frac{1}{1-x^2}} \). A simpler fraction, easier to interpret.

These steps showcase the importance of fraction manipulation and simplification in the broader scope of algebraic expressions, aiding in arriving at the simplest and most interpretable form of the given expression.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying algebraic expressions using various algebraic rules and properties. This skill is vital for making complex problems more manageable, as it involves strategic operations to reveal simpler forms of a problem.

• **Order of Operations**: It's crucial to follow the correct order while simplifying. Prioritize parentheses, exponents, multiplication, division, addition, and subtraction (PEMDAS/BODMAS).
• **Distributive Property**: This property allows multiplying across terms inside parentheses and was utilized when we combined terms inside the square root to form a new fraction.
• **Fraction Operations**: Simplifying across fractions by finding common denominators or, in this case, by simplifying \( \sqrt{\frac{1}{1-x^2}} \) to further simplify to \( \frac{1}{\sqrt{1-x^2}} \).

In the exercise, after substituting the simplified fraction back into the main expression and observing the cancellations, algebraic manipulation was key in handling any further simplifications required to reach the final simplest form. This method ensures the expression is not only simplified but maintains its original properties and value, critical for accurate problem-solving in mathematics.