Problem 4 - Algebra 2 and Trigonometry

Question

Rationalize the denominator and write each fraction in simplest form. All variables represent positive numbers. \(\frac{5}{\sqrt{10}}\)

Step-by-Step Solution

Answer

The fraction \( \frac{5}{\sqrt{10}} \) simplifies to \( \frac{\sqrt{10}}{2} \).

Step 1: Identify the problem

We need to rationalize the denominator of \( \frac{5}{\sqrt{10}} \) and simplify it. Rationalizing the denominator involves eliminating the square root in the denominator.

Step 2: Multiply by a form of 1

To eliminate the square root in the denominator, multiply the numerator and the denominator by \( \sqrt{10} \). This step is equivalent to multiplying the entire fraction by 1, which does not change its value.\[ \frac{5}{\sqrt{10}} \times \frac{\sqrt{10}}{\sqrt{10}} = \frac{5 \cdot \sqrt{10}}{\sqrt{10} \cdot \sqrt{10}} \]

Step 3: Simplify the Denominator

The denominator is \( \sqrt{10} \cdot \sqrt{10} = 10 \) as squaring a square root results in the number itself.\[ \frac{5 \cdot \sqrt{10}}{10} \]

Step 4: Simplify the Fraction

To simplify, divide both the numerator and the denominator by their greatest common divisor, which is 5.\[ \frac{5 \cdot \sqrt{10}}{10} = \frac{\sqrt{10}}{2} \]

Step 5: Write the Final Answer

After simplifying, the rationalized fraction is \( \frac{\sqrt{10}}{2} \). This fraction is in its simplest form with a rationalized denominator.

Key Concepts

Simplifying Fractions

Simplifying fractions is an essential skill in mathematics. The objective is to reduce a fraction to its simplest form. This means the numerator and denominator have no common factors other than 1. Simplifying fractions makes problems easier to understand and solve.

To simplify a fraction, we look for the greatest common divisor (GCD) of both the numerator and the denominator. Once identified, we divide them by this GCD:

• This reduces the fraction to its simplest form.
• Essentially, simplifying fractions helps in comparison and further calculations.

In the solution, after rationalizing the denominator, we simplified \( \frac{5 \cdot \sqrt{10}}{10} \) by dividing both the numerator and denominator by 5, the GCD, resulting in \( \frac{\sqrt{10}}{2} \). This step not only makes it simpler but also easier to interpret in further operations.

Square Roots

Square roots are a fundamental concept in mathematics. A square root of a number is a value that, when multiplied by itself, gives the original number.

Let's consider the number 10. The square root of 10 is represented as \( \sqrt{10} \). While this is an irrational number, it's crucial when working with fractions or irrational numbers to know how they behave:

• Multiplying \( \sqrt{10} \) by itself yields 10, as seen in rationalizing denominators.
• This squaring of square roots helps in rationalizing denominators and simplifying expressions.

In simplifying \( \frac{5}{\sqrt{10}} \), we eliminated the square root from the denominator by multiplying the fraction by \( \frac{\sqrt{10}}{\sqrt{10}} \), which is tantamount to removing the square root. This is a key step in any fraction containing square roots.

Greatest Common Divisor

The greatest common divisor (GCD) of two numbers is the largest number that divides both of them without leaving a remainder. Knowing how to find the GCD is practical for simplifying fractions:

• Identify the common factors of the numerator and the denominator.
• The GCD is the largest of these factors. Dividing both parts of the fraction by the GCD reduces it to lowest terms.

In our example, after rationalizing the denominator, we had \( \frac{5 \cdot \sqrt{10}}{10} \). Recognizing that 5 is the GCD of 5 and 10, we divided both terms by their GCD, simplifying the fraction effectively to \( \frac{\sqrt{10}}{2} \). Understanding and applying the GCD is crucial for simplifying fractions, thus making them more manageable and interpretable.