Problem 16 - A Graphical Approach to Precalculus with Limits

Question

Rationalize the denominators for the given expressions. Assume all expressions containing \(x\) are positive. $$\frac{7}{\sqrt{4-x^{2}}}$$

Step-by-Step Solution

Answer

\( \frac{7\sqrt{4-x^2}}{4-x^2} \)

Step 1: Identify the Goal

The goal is to rationalize the denominator of the expression \( \frac{7}{\sqrt{4-x^2}} \). This means we need to eliminate the square root from the denominator.

Step 2: Multiply by the Conjugate

To rationalize the denominator, multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of \( \sqrt{4-x^2} \) is itself, since it is a simple radical expression without other terms. Thus, multiply by \( \frac{\sqrt{4-x^2}}{\sqrt{4-x^2}} \).

Step 3: Simplify the Resulting Expression

When you multiply the numerator and the denominator by \( \sqrt{4-x^2} \), the expression becomes:\[\frac{7 \cdot \sqrt{4-x^2}}{(\sqrt{4-x^2})^2}\]In the denominator, \( (\sqrt{4-x^2})^2 = 4-x^2 \).

Step 4: Final Simplified Expression

The expression is now:\[\frac{7\sqrt{4-x^2}}{4-x^2}\]This is the rationalized form of the original expression, as the denominator no longer contains a square root.

Key Concepts

Radical Expressions

Radical expressions include roots of numbers or variables, commonly square roots or cube roots. Understanding how to work with them is crucial because they frequently pop up in algebra and calculus.
Square roots are the most common radical expressions you'll encounter. They're denoted by the radical symbol \( \sqrt{} \). For example, \( \sqrt{9} \) equals 3 because 3 multiplied by itself gives 9. A radical expression can also contain variables, as seen in \( \sqrt{4-x^2} \).

• To manipulate radical expressions, knowing how to add, subtract, multiply, or divide them is essential. For instance, square roots can be simplified if the number under the root is a perfect square.
• If you have \( \sqrt{b^2} \), you can simplify it to \( b \) assuming \( b \) is non-negative.

Radical expressions in a denominator can complicate an equation, making it harder to perform arithmetic operations. That's why rationalizing the denominator, or removing the radical from it, is so valuable in mathematics.

Conjugate

The conjugate of a radical expression is a tool used to eliminate radicals from the denominator of a fraction. It's a particularly useful concept when simplifying expressions.
For simple radical expressions like \( \sqrt{4-x^2} \), the conjugate is the expression itself. However, if the radical expression is part of a binomial, such as \( a + \sqrt{b} \), its conjugate would be \( a - \sqrt{b} \).
Using the conjugate effectively helps rationalize the denominator:

• Multiply both the numerator and the denominator by the conjugate of the denominator.
• For binomials \( a + \sqrt{b} \), multiplying by its conjugate \( a - \sqrt{b} \) results in a difference of squares, \( a^2 - b \), removing the square root.

In our exercise, since the denominator \( \sqrt{4-x^2} \) is a simple radical, its conjugate remains \( \sqrt{4-x^2} \).

Simplifying Expressions

The process of simplifying expressions involves making them easier to read and work with, without changing their value. It's vital for solving equations efficiently.
When you simplify a fraction, you're looking to make both the numerator and the denominator as simple as possible. In our example, the goal was to eliminate the square root from the denominator by multiplying by the conjugate.

• The multiplication of the denominator \( \sqrt{4-x^2} \) by its conjugate \( \sqrt{4-x^2} \) yields \( (\sqrt{4-x^2})^2 \), simplifying to \( 4-x^2 \).
• This results in an expression where the denominator is now a polynomial \( 4-x^2 \), without radicals.

Mastering simplification techniques helps tackle complex calculations and improve problem-solving skills in mathematics. As you practice, these approaches become intuitive, quickening your mathematical reasoning and efficiency.