Problem 349 - Elementary Algebra

Question

In the following exercises, simplify and rationalize the denominator. $$ \frac{3}{\sqrt{13}} $$

Step-by-Step Solution

Answer

\( \frac{3 \sqrt{13}}{13} \)

Step 1 - Multiply the numerator and denominator by the same radical

To rationalize the denominator, multiply both the numerator and the denominator by \(\sqrt{13}\) to eliminate the radical in the denominator. The expression becomes: \[ \frac{3}{\sqrt{13}} \times \frac{\sqrt{13}}{\sqrt{13}} = \frac{3 \cdot \sqrt{13}}{\sqrt{13} \cdot \sqrt{13}} \]

Step 2 - Simplify the denominator

Simplify the denominator by using the property of radicals \(\sqrt{a} \cdot \sqrt{a} = a\). This gives us: \[ \frac{3 \sqrt{13}}{13} \]

Step 3 - Final simplified form

The expression cannot be simplified any further. The fully simplified and rationalized form is: \[ \frac{3 \sqrt{13}}{13} \]

Key Concepts

Simplify Radicals

Simplifying radicals involves rewriting a radical expression in its simplest form.
Radicals, like square roots, are simplified by ensuring that there are no perfect square factors other than 1 inside the radical. For example, √18 can be simplified to 3√2 because 18 = 9 × 2 and √9 = 3.
In our exercise, the radical \( \sqrt{13} \) is already in its simplest form because 13 is a prime number. Prime numbers cannot be broken down into smaller factors, so there's no way to simplify \( \sqrt{13} \) any further.
Remember, simplifying radicals makes it easier to work with them in equations and other mathematical expressions.

Rationalizing Denominators

Rationalizing the denominator is the process of eliminating the radical from the denominator of a fraction.
To do this, we multiply both the numerator and the denominator by the radical found in the denominator. By doing this, the radical is effectively 'moved' to the numerator. For example, in our exercise: \[ \frac{3}{\sqrt{13}} \]
We multiply both the numerator and the denominator by \( \sqrt{13} \): \[ \frac{3 \cdot \sqrt{13}}{\sqrt{13} \cdot \sqrt{13}} = \frac{3 \sqrt{13}}{13} \]
This works because multiplying \( \sqrt{13} \) by itself results in 13, which is a rational number.
This step is essential as it clears the radical from the denominator, making the fraction easier to handle in further calculations or in more complex mathematical contexts.

Simplifying Fractions

Once a fraction's denominator has been rationalized, the next step is to simplify the fraction if possible.
Simplifying a fraction means reducing it to its simplest form where the numerator and the denominator have no common factors other than 1.
For instance, in our given problem, after rationalizing, we get: \[ \frac{3 \sqrt{13}}{13} \]
In this case, the fraction \( \frac{3}{13} \) cannot be simplified any further since 3 and 13 share no common factors. Therefore, \( \frac{3 \sqrt{13}}{13} \) is in its simplest form.
Simplifying fractions helps reduce the complexity of the expression, making it cleaner and easier to understand and use in further mathematical operations.