Problem 354 - Elementary Algebra

Question

In the following exercises, simplify and rationalize the denominator. $$ \frac{9}{2 \sqrt{7}} $$

Step-by-Step Solution

Answer

\( \frac{9 \sqrt{7}}{14} \)

Step 1: Identify the problem

The given expression is \(\frac{9}{2 \, \sqrt{7}}\). The goal is to simplify and rationalize the denominator.

Step 2: Multiply by the conjugate

To rationalize the denominator, multiply both the numerator and the denominator by \(\sqrt{7}\). This gives: \[ \frac{9}{2 \, \sqrt{7}} \times \frac{\sqrt{7}}{\sqrt{7}} \]

Step 3: Simplify the multiplication

Simplify the multiplication in both the numerator and the denominator: \[ \frac{9 \sqrt{7}}{2 \, \sqrt{7} \, \sqrt{7}} \]

Step 4: Simplify the denominator

Since \( \sqrt{7} \, \sqrt{7} = 7 \), the expression becomes:\[ \frac{9 \sqrt{7}}{2 \, 7} \]

Step 5: Final simplification

Simplify the fraction: \[ \frac{9 \sqrt{7}}{14} \]

Key Concepts

Rationalizing the Denominator

When you have a fraction with a square root in the denominator, like \(\frac{9}{2 \sqrt{7}}\), you need to rationalize it. This means removing the square root from the denominator. To do this, multiply both the numerator and the denominator by the same square root present in the denominator. This ensures the value of the fraction remains unchanged. For the given example, we multiply both parts by \(\sqrt{7}\), resulting in \(\frac{9 \sqrt{7}}{2 \sqrt{7} \sqrt{7}}\). Simplifying further will lead to a more presentable form without any radicals in the denominator.

Simplifying Fractions

Simplifying fractions involves reducing them to their simplest form. Start by multiplying out the radicals in the denominator. For example, \(2 \sqrt{7} \sqrt{7} = 2 \times 7 = 14\). Thus, the fraction becomes \(\frac{9 \sqrt{7}}{14}\). The process of simplifying fractions is essential to make the expression cleaner and easier to work with in future calculations.
Ensure you look for common factors that can cancel out from the numerator and the denominator to simplify further. In this case, there are no more common factors, and the fraction \(\frac{9 \sqrt{7}}{14}\) is fully simplified.

Multiplying Radicals

When multiplying radicals, it’s essential to remember that \(\sqrt{a} \times \sqrt{b} = \sqrt{a \times b}\). In our specific example, \(2 \sqrt{7} \sqrt{7}\) simplifies to \(2 \sqrt{7 \times 7} = 2 \sqrt{49} = 2 \times 7 = 14\). This step often involves simplifying the multiplication inside the radicals first, followed by any numerical multiplication outside of the radicals. Multiplying radicals is a crucial skill in algebra, allowing easier manipulation and simplification of expressions.

Algebraic Steps

Each algebraic step in simplifying \(\frac{9}{2 \sqrt{7}}\) is crucial. First, identify the need to rationalize the denominator. Second, correctly multiply by \(\sqrt{7}\) to get \(\frac{9 \sqrt{7}}{2 \sqrt{7} \sqrt{7}}\). Third, simplify the multiplication to rewrite the denominator as \(14\). Finally, ensure the fraction is in its simplest form by checking for any common factors to cancel out. These steps illustrate the importance of systematic and detailed thinking in algebra to reach the correct, simplified solution.