Problem 351 - Elementary Algebra

Question

In the following exercises, simplify and rationalize the denominator. $$ \frac{10}{3 \sqrt{10}} $$

Step-by-Step Solution

Answer

The simplified form is \( \frac{\sqrt{10}}{3} \).

Step 1: Identify the Problem

We need to simplify the expression \( \frac{10}{3 \sqrt{10}} \) and rationalize the denominator.

Step 2: Multiply Numerator and Denominator by \( \sqrt{10} \)

To rationalize the denominator, multiply both the numerator and the denominator by \( \sqrt{10} \). This gives: \[ \frac{10}{3 \sqrt{10}} \times \frac{\sqrt{10}}{\sqrt{10}} = \frac{10 \sqrt{10}}{3 \sqrt{10} \sqrt{10}} \].

Step 3: Simplify the Denominator

Simplify the denominator using the fact that \( \sqrt{10} \cdot \sqrt{10} = 10 \). This gives: \[ \frac{10 \sqrt{10}}{3 \times 10} = \frac{10 \sqrt{10}}{30} \].

Step 4: Simplify the Whole Fraction

Divide both the numerator and the denominator by 10 to simplify: \[ \frac{10 \sqrt{10}}{30} = \frac{\sqrt{10}}{3} \].

Key Concepts

Simplification

Simplification is the process of reducing an expression to its simplest form. This could involve combining like terms, reducing fractions, or eliminating radicals in the denominator of a fraction. By simplifying, we make expressions easier to understand and work with.

In the given exercise, we started with the fraction \( \frac{10}{3 \, \text{sqrt}(10)} \). Our goal was to simplify this fraction as much as possible. Here's a quick overview of the steps:

• Identify the problem and determine what needs to be simplified.
• Multiply the numerator and the denominator by the same value (in this case, \( \text{sqrt}(10) \)), to rationalize the denominator.
• Simplify the fraction by combining like terms or reducing the fraction if possible.

This method ensured the fraction was reduced to its simplest form.

Radicals

Radicals involve expressions containing roots, such as square roots or cube roots. In our example, we dealt with a square root: \( \text{sqrt}(10) \). Radicals can sometimes make it difficult to simplify or use fractions, especially if they are in the denominator.

When we say 'rationalizing the denominator,' we're essentially trying to eliminate these radicals from the bottom part of the fraction. We do this by multiplying both the numerator and the denominator by a value that will help remove the radical.

In our exercise, this value was \( \text{sqrt}(10) \). By multiplying both top and bottom by \( \text{sqrt}(10) \), we transformed \( 3 \, \text{sqrt}(10) \) into \( 30 \), a rational number.

Fractions

Fractions consist of a numerator (top part) and a denominator (bottom part) separated by a division line. Simplifying fractions often involves finding common factors or using mathematical operations to make both parts simpler.

In the initial fraction \( \frac{10}{3 \, \text{sqrt}(10)} \), our denominator contained a radical. To deal with this, we multiplied by \( \text{sqrt}(10) \), giving us a fraction with a rational denominator: \( \frac{10 \, \text{sqrt}(10)}{30} \).

The next task was to simplify the entire fraction by reducing it. By dividing both the numerator and denominator by their greatest common divisor (in this case, 10), we arrived at the final simplified form: \( \frac{\text{sqrt}(10)}{3} \). This step is crucial for making fractions more manageable and easier to understand.