Problem 10 - Algebra 2 and Trigonometry

Question

Rationalize the denominator and write each fraction in simplest form. All variables represent positive numbers. \(\frac{6}{\sqrt{12}}\)

Step-by-Step Solution

Answer

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

Step 1: Identify the Problem

We have to rationalize the denominator of the fraction \( \frac{6}{\sqrt{12}} \). This means we will get rid of the square root in the denominator.

Step 2: Simplify the Denominator

First, simplify \( \sqrt{12} \) by expressing 12 as a product of its prime factors: \( 12 = 4 \times 3 \). So, \( \sqrt{12} = \sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3} = 2 \sqrt{3} \). Therefore, \( \frac{6}{\sqrt{12}} = \frac{6}{2\sqrt{3}} \).

Step 3: Simplify the Fraction

Simplify the fraction \( \frac{6}{2\sqrt{3}} \): divide both the numerator and the non-root part of the denominator by 2 to get \( \frac{3}{\sqrt{3}} \).

Step 4: Rationalize the Denominator

To remove the square root from the denominator, multiply the numerator and denominator by \( \sqrt{3} \). So, \( \frac{3}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{3\sqrt{3}}{3} \).

Step 5: Simplify Further

Finally, simplify \( \frac{3\sqrt{3}}{3} \), which equals \( \sqrt{3} \) since the 3s in the numerator and denominator cancel each other out.

Key Concepts

Simplifying Radicals

When working with radicals, especially within fractions, it's crucial to simplify them as much as possible. Simplifying radicals makes further operations, like rationalizing the denominator, much easier. A radical is simplified when no perfect squares (other than 1) are left within the square root.

Consider the fraction \( \frac{6}{\sqrt{12}} \). To simplify \( \sqrt{12} \), recognize that 12 can be expressed as a product of numbers where one is a perfect square. Here, \( 12 = 4 \times 3 \). By breaking it down into prime factors, we find \( 4 \times 3 \), where 4 is a perfect square equal to \( 2^2 \). The radical can then be simplified:

• \( \sqrt{12} = \sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3} \)
• \( \sqrt{4} = 2 \)

Thus, \( \sqrt{12} = 2\sqrt{3} \). This simplification process is crucial to handle more complex mathematical expressions easily and efficiently.

Simplifying radicals not only helps in solving exercises but also strengthens understanding of the underlying mathematical principles involved.

Prime Factorization

Prime factorization is the process of breaking down a number into its prime number components. This technique is particularly useful in simplifying radicals and fractions.

Let's take 12 as an example, which appears in the denominator of our fraction \( \frac{6}{\sqrt{12}} \). The prime factorization of 12 is the step where 12 is expressed as a product of its prime factors:

• Start by finding two numbers that multiply to give 12, say 4 and 3.
• 4 is not a prime number, so break it down further: \( 4 = 2 \times 2 \).
• This gives us: \( 12 = 2 \times 2 \times 3 \).

Prime factorization simplifies not just radicals but can also be used to simplify fractions by revealing common factors.

By understanding prime factorization, you can readily simplify expressions and better tackle problems that include variable components as well.

Simplifying Fractions

Simplifying fractions is an essential skill in mathematics that ensures expressions are in their simplest form. This involves dividing both the numerator and the denominator by their greatest common factor (GCF).

Returning to our example, \( \frac{6}{2\sqrt{3}} \), we know that the numerator and the non-radical part of the denominator, 2, share a common factor. By dividing both by 2, we can simplify the fraction:

• \( \frac{6}{2\sqrt{3}} = \frac{6 \div 2}{2\sqrt{3} \div 2} = \frac{3}{\sqrt{3}} \).

Now, to further simplify this fraction and rationalize the denominator, we multiply both the numerator and the denominator by \( \sqrt{3} \):

• \( \frac{3}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{3\sqrt{3}}{3} \), which simplifies to \( \sqrt{3} \).

The focus is not only on simplifying numbers but also on ensuring that the final expression is more accessible and easier to work with. Simplifying fractions is a fundamental tool that leads to clearer, more concise mathematical solutions.