Problem 15 - A Graphical Approach to Precalculus with Limits

Question

Rationalize the denominators for the given expressions. Assume all expressions containing \(x\) are positive. $$\frac{6}{\sqrt{1-x^{2}}}$$

Step-by-Step Solution

Answer

The rationalized form is \( \frac{6\sqrt{1-x^2}}{1-x^2} \).

Step 1: Identify the Problem

The given expression is \( \frac{6}{\sqrt{1-x^2}} \). The goal is to rationalize the denominator, which means to remove the square root from the denominator.

Step 2: Multiply by Conjugate

To eliminate the square root from the denominator, multiply both the numerator and the denominator by \( \sqrt{1-x^2} \) (the conjugate of the denominator). This gives us: \( \frac{6}{\sqrt{1-x^2}} \times \frac{\sqrt{1-x^2}}{\sqrt{1-x^2}} = \frac{6\sqrt{1-x^2}}{1-x^2} \).

Step 3: Simplify the Expression

The denominator is now \(1-x^2\) because \( \sqrt{1-x^2} \times \sqrt{1-x^2} = 1-x^2 \). The expression becomes \( \frac{6\sqrt{1-x^2}}{1-x^2} \).

Key Concepts

Square Root

The square root is a mathematical function symbolized by the radical sign \( \sqrt{} \). It represents a number which when multiplied by itself gives the original number. For instance, \( \sqrt{4} = 2 \) because \( 2 \times 2 = 4 \). Similarly, \( \sqrt{9} = 3 \) because \( 3 \times 3 = 9 \). Square roots are common in mathematics, including calculus and algebra. Often they appear in complex equations, in which they may need to be manipulated for simplification or solving purposes.

• The square root of a positive number is always non-negative.
• In the context of rationalizing the denominator, if you see a square root in the denominator like in \( \sqrt{1-x^2}\), it is usually removed by a specific technique.

In mathematics, we often try to eliminate square roots from the denominator of fractions to simplify the expression, making it more useful for further calculations.

Conjugate

The conjugate in mathematics refers to a binomial formed by changing the sign between two terms. For example, the conjugate of \( \sqrt{a} + b \) is \( \sqrt{a} - b \). In rationalizing denominators, using the conjugate is a crucial technique.When the denominator involves a square root, such as \( \sqrt{1-x^2} \), multiplying by the conjugate helps in removing the square root. The reason this works is that multiplying a binomial by its conjugate results in a difference of squares.For instance:

• Original binomial: \( \sqrt{1-x^2} \)
• Conjugate: \( \sqrt{1-x^2} \)
• Formula: \( (\sqrt{1-x^2})(\sqrt{1-x^2}) = 1-x^2 \)

By using the conjugate, we achieve a rational denominator which simplifies further mathematical processes. It ensures that the expression is easier to interpret and can be more conveniently evaluated in equations.

Simplify the Expression

Simplifying an expression means to rewrite it in the most concise and efficient form. Simplification can involve several processes, including combining like terms, reducing fractions, and eliminating any unnecessary components.In our original exercise, after multiplying by the conjugate, we attained \( \frac{6\sqrt{1-x^2}}{1-x^2} \). Here:

• The numerator has \( 6\sqrt{1-x^2} \), combining the constant 6 and the square root \( \sqrt{1-x^2} \).
• The denominator is \( 1-x^2 \), a significant improvement as the square root is removed.

By simplifying the expression, you express it in a way that's straightforward to handle in further algebraic operations or calculus formulas.Remember:

• Simplification doesn't change the inherent value of the expression.
• It often involves reducing fractions or performing operations to provide a cleaner result.

This step ensures that the expressions used in calculations are optimal, reducing potential errors and making mathematical problem-solving more effective and efficient.