Problem 69 - Algebra and Trigonometry

Question

\(65-70\) m 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: Rewrite the Expression

The expression inside the square root is \(1+\left(\frac{x}{\sqrt{1-x^{2}}}\right)^{2}\). Let's work on simplifying \(\left(\frac{x}{\sqrt{1-x^2}}\right)^2\) first.

Step 2: Simplify the Fractional Part

Calculate \(\left(\frac{x}{\sqrt{1-x^2}}\right)^2\):\[\left(\frac{x}{\sqrt{1-x^2}}\right)^2 = \frac{x^2}{1-x^2}\]

Step 3: Combine Terms Inside the Square Root

Now combine the terms inside the square root:\[1 + \frac{x^2}{1-x^2} = \frac{(1-x^2) + x^2}{1-x^2} = \frac{1-x^2+x^2}{1-x^2} = \frac{1}{1-x^2}\]

Step 4: Simplify the Entire Expression

Now simplify the square root of the expression:\[\sqrt{\frac{1}{1-x^2}} = \frac{1}{\sqrt{1-x^2}}\]

Key Concepts

Simplification

Simplification is a critical process in mathematics that involves reducing an expression to its most basic form without changing its overall value. This makes the expression more manageable and easier to understand.

In the given problem, we aim to simplify a fractional expression. Initially, we have a complex expression under a square root, which includes a fraction raised to a power. To simplify:

• First, square the fraction. This means, in our problem, taking \( \left( \frac{x}{\sqrt{1-x^2}} \right)^2 \) and simplifying it to \( \frac{x^2}{1-x^2} \).
• Next, integrate the resultant fraction into the rest of the expression. Here, the expression becomes \[ 1 + \frac{x^2}{1-x^2} = \frac{1-x^2+x^2}{1-x^2} = \frac{1}{1-x^2} \]. By performing this operation, we effectively streamline the expression.

Simplification tackles potential complexity by systematically breaking down the components of an expression, ensuring maximum clarity and ease of calculation.

Square Root

A square root of a number or expression is a value that, when multiplied by itself, gives the original number or expression. Understanding square roots is essential in algebra, as it helps deal with powers and radical expressions.

In our problem, the expression involves simplifying under a square root. The expression becomes \( \sqrt{\frac{1}{1-x^2}} \).

• The square root symbol \( \sqrt{} \) indicates what you need to find such that squaring the result gives you the original number or expression.
• To simplify this expression, remember that \( \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \).
• Applying the rule here simplifies it to \( \frac{1}{\sqrt{1-x^2}} \), using the identity that \( \sqrt{1} = 1 \).

Mastering square roots involves familiarization with such properties, making radical expressions easier to manipulate.

Rationalization

Rationalization is the process of eliminating square roots or radicals from the denominator of a fraction. This is standard practice because it makes further algebraic operations more straightforward and your expressions more presentable.

In simpler terms, it means transforming a fractional expression so that no roots appear in its denominator. While the expression \( \frac{1}{\sqrt{1-x^2}} \) we derived in the solution isn't rationalized, rationalizing such expressions typically involves:

• Multiplying the numerator and the denominator by the square root found in the denominator.
• For example, if we have \( \frac{1}{\sqrt{b}} \,\) you would multiply top and bottom by \( \sqrt{b} \,\) to yield \( \frac{\sqrt{b}}{b} \).
• This removes the radical from the denominator while preserving the equivalence of the expression.

Rationalization ensures expressions are in a form that is generally preferred in mathematical writing, making them cleaner and often easier to work with in further calculations.