Problem 12 - A Graphical Approach to Precalculus with Limits

Question

Rationalize the denominators for the given expressions. Assume all expressions containing \(x\) are positive. $$\frac{1}{3 \sqrt{7}}$$

Step-by-Step Solution

Answer

The rationalized expression is \( \frac{\sqrt{7}}{21} \).

Step 1: Identify the Expression

We start with the expression given in the problem: \( \frac{1}{3 \sqrt{7}} \). The issue here is the square root in the denominator.

Step 2: Multiply by the Conjugate

To rationalize the denominator, multiply both the numerator and denominator by \( \sqrt{7} \). This will help eliminate the square root from the denominator.

Step 3: Perform the Multiplication

Multiply the numerator and the denominator by \( \sqrt{7} \). The expression becomes \( \frac{1 \times \sqrt{7}}{3 \sqrt{7} \times \sqrt{7}} = \frac{\sqrt{7}}{3 \cdot 7} \).

Step 4: Simplify the Denominator

Simplify the denominator, as \(3 \cdot 7 = 21\). Hence, the expression simplifies to \( \frac{\sqrt{7}}{21} \).

Key Concepts

Square Roots

Square roots are fundamental mathematical expressions represented as \( \sqrt{x} \), where \( x \) is a non-negative number. The square root of a number is a value that, when multiplied by itself, gives the original number. For example, \( \sqrt{9} = 3 \) because \( 3 \times 3 = 9 \).

In operations involving rationalizing denominators, square roots can often appear in fractions, such as \( \frac{1}{3\sqrt{7}} \). In these cases, the goal is to "rationalize" or convert the denominator to a rational number, free from square roots, while maintaining the equality of the expression.

To effectively handle square roots and perform operations on them, remember:

• \( \sqrt{a} \times \sqrt{a} = a \)
• Only non-negative numbers have real square roots.

These rules are very useful when simplifying fractions with square roots in the denominator, as seen in solving rationalization problems.

Conjugate Multiplication

Conjugate multiplication is a technique used to eliminate irrational numbers, often square roots, from the denominator of a fraction. This method involves multiplying both the numerator and the denominator by a specific expression known as the conjugate.

In the case of the expression \( \frac{1}{3\sqrt{7}} \), the conjugate of \( \sqrt{7} \) is simply itself \( \sqrt{7} \) because it does not have any terms to change signs. By multiplying both the numerator and the denominator by \( \sqrt{7} \), we can eliminate the square root:

\[ \frac{1}{3\sqrt{7}} \times \frac{\sqrt{7}}{\sqrt{7}} = \frac{\sqrt{7}}{3\sqrt{7}\times\sqrt{7}} = \frac{\sqrt{7}}{21} \]

This process simplifies the denominator as \( \sqrt{7} \times \sqrt{7} = 7 \), which results in a more manageable fraction. It is essential to make sure that each step maintains the equality of the equation by multiplying "top and bottom" by the same term.

Simplifying Expressions

Simplifying expressions means reducing them to their simplest, most compact form without changing their value. This often includes combining like terms, reducing fractions, and removing square roots from denominators.

In our example, after multiplying by the conjugate, we obtained \( \frac{\sqrt{7}}{21} \). The denominator's simplification from \( 3 \times 7 \) to 21 makes it rational and easier to understand.

• Always look for common factors to reduce fractions.
• Ensure the equality of expressions by performing identical operations on both sides of the fraction line.

This ensures clarity and ease when performing further calculations or integrating the expression into more complex scenarios.

By mastering the simplification process, you make mathematical problems easier to approach and solve.