Problem 196 - Elementary Algebra

Question

In the following exercises, simplify. $$ \frac{1}{3} \sqrt{24}+\frac{1}{4} \sqrt{54} $$

Step-by-Step Solution

Answer

\( \frac{17 \sqrt{6}}{12} \)

Step 1 - Simplify the Radicals

Start by simplifying the radicals. Notice that \( \sqrt{24} \) can be written as \( \sqrt{4 \cdot 6} = \sqrt{4} \cdot \sqrt{6} = 2 \sqrt{6} \). Similarly, \( \sqrt{54} \) can be written as \( \sqrt{9 \cdot 6} = \sqrt{9} \cdot \sqrt{6} = 3 \sqrt{6} \).

Step 2 - Substitute Simplified Radicals

Substitute the simplified radicals back into the expression: \( \frac{1}{3} \cdot 2 \sqrt{6} + \frac{1}{4} \cdot 3 \sqrt{6} \).

Step 3 - Multiply the Coefficients

Multiply the coefficients to simplify each term: \( \frac{1}{3} \cdot 2 \sqrt{6} = \frac{2 \sqrt{6}}{3} \) and \( \frac{1}{4} \cdot 3 \sqrt{6} = \frac{3 \sqrt{6}}{4} \).

Step 4 - Find a Common Denominator

To combine the terms, find a common denominator. The common denominator for 3 and 4 is 12. Rewrite each fraction with this common denominator: \( \frac{2 \sqrt{6}}{3} = \frac{8 \sqrt{6}}{12} \) and \( \frac{3 \sqrt{6}}{4} = \frac{9 \sqrt{6}}{12} \).

Step 5 - Combine the Terms

Add the fractions together by combining the numerators: \( \frac{8 \sqrt{6}}{12} + \frac{9 \sqrt{6}}{12} = \frac{17 \sqrt{6}}{12} \). The final simplified form is \( \frac{17 \sqrt{6}}{12} \).

Key Concepts

Understanding Radical Expressions

A radical expression involves a root, most commonly the square root. In our problem, we start with the square roots of 24 and 54. To simplify these radicals, we break down each number into its prime factors. For example, 24 can be written as \( 24 = 4 \times 6 \), where 4 is a perfect square. Taking the square root of 4 gives us 2, making the simplified form \( 2 \times \sqrt{6} \). Similarly, 54 can be broken down into \( 9 \times 6 \). The square root of 9 is 3, hence simplifying \( \sqrt{54} \) to \( 3 \sqrt{6} \).

Simplifying radical expressions helps in making further mathematical operations, like addition and subtraction, more straightforward. Always look for perfect squares within the radical that can be simplified.

Finding Common Denominators

When dealing with fractions, a common denominator is essential for addition or subtraction. A common denominator is a shared multiple of the denominators of the fractions involved. In our problem, after simplifying the radicals, we are left with fractions \( \frac{2 \sqrt{6}}{3} \) and \( \frac{3 \sqrt{6}}{4} \).

To add these, we need the same denominator. The least common multiple (LCM) of 3 and 4 is 12. Converting each fraction:

• For \( \frac{2 \sqrt{6}}{3} \), multiply the numerator and the denominator by 4 to get \( \frac{8 \sqrt{6}}{12} \)
• For \( \frac{3 \sqrt{6}}{4} \), multiply the numerator and the denominator by 3 to get \( \frac{9 \sqrt{6}}{12} \)

Now both fractions share a common denominator of 12, making them ready to be added together.

Adding Fractions

Once you've found a common denominator, adding fractions becomes a matter of combining the numerators while keeping the denominator the same. From our previous steps, we have:

• \( \frac{8 \sqrt{6}}{12} \)
• \( \frac{9 \sqrt{6}}{12} \)

Sum these fractions by adding their numerators: \( 8 \sqrt{6} + 9 \sqrt{6} = 17 \sqrt{6} \). Keeping the common denominator, the final result is \( \frac{17 \sqrt{6}}{12} \).

Here’s key advice: always check that the denominators match before adding or subtracting fractions. This ensures the operation is mathematically sound and simplifies the process significantly.