Problem 80 - College Algebra with Modeling and Visualization

Question

Simplify the expression. Assume that all variables are positive. $$ \frac{23 \sqrt{11}}{2}-\frac{\sqrt{44}}{8} $$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{45 \sqrt{11}}{4} \).

Step 1: Simplify the Second Term

First, simplify the square root in the second term. Observe that \( \sqrt{44} = \sqrt{4 \times 11} \). Since \( \sqrt{4} = 2 \), this simplifies to \( 2 \sqrt{11} \). Hence, the expression becomes \( \frac{23 \sqrt{11}}{2} - \frac{2 \sqrt{11}}{8} \).

Step 2: Simplify the Fraction in the Second Term

Next, simplify the fraction \( \frac{2 \sqrt{11}}{8} \). We can divide both the numerator and the denominator by 2 to get \( \frac{\sqrt{11}}{4} \). So the expression now is \( \frac{23 \sqrt{11}}{2} - \frac{\sqrt{11}}{4} \).

Step 3: Find a Common Denominator

In order to subtract the fractions, we need a common denominator. The least common multiple of 2 and 4 is 4. Convert \( \frac{23 \sqrt{11}}{2} \) to have the denominator 4: \( \frac{23 \sqrt{11} \times 2}{2 \times 2} = \frac{46 \sqrt{11}}{4} \).

Step 4: Subtract the Fractions

Now that both terms have the same denominator, subtract the fractions: \( \frac{46 \sqrt{11}}{4} - \frac{\sqrt{11}}{4} \). This results in \( \frac{(46 - 1) \sqrt{11}}{4} = \frac{45 \sqrt{11}}{4} \).

Step 5: Write the Simplified Expression

Thus, the simplified expression is \( \frac{45 \sqrt{11}}{4} \).

Key Concepts

Square Roots

Square roots are a fundamental concept in mathematics that find the number that, when multiplied by itself, gives the original number. In this exercise, you see square roots in action with expressions like \( \sqrt{44} \). This can be broken down further to \( \sqrt{4 \times 11} \). By knowing that \( \sqrt{4} = 2 \), we simplify \( \sqrt{44} \) to \( 2\sqrt{11} \).

• Break down numbers into factors to simplify square roots.
• Remember that square roots applied to perfect squares can be simplified easily.

Understanding and simplifying square roots can often make algebraic expressions much easier to work with, as shown in this simplification process.

Fractions

Fractions represent parts of a whole and are denoted with a numerator and a denominator. In algebra, fractions appear regularly, often involving variables under the square root or multiplied by other terms. In the given problem, the first step involves the fraction \( \frac{2\sqrt{11}}{8} \), which was later simplified by factoring.

• A fraction’s numerator is the top number, and the denominator is the bottom number.
• To simplify, divide both the numerator and denominator by their greatest common factor.

By simplifying fractions properly, you can make your algebraic expressions simpler and more manageable.

Common Denominator

To subtract fractions effectively, they must share a common denominator. This ensures you compare and combine like terms accurately. For example, in the exercise, the fractions \( \frac{23 \sqrt{11}}{2} \) and \( \frac{\sqrt{11}}{4} \) are aligned with a common denominator of 4.

• Find the least common multiple (LCM) of the denominators.
• Adjust each fraction so that they have the same denominator.

This step helps simplify complex fraction operations and makes subtraction straightforward, allowing for cleaner solutions.

Subtraction of Fractions

Subtracting fractions follows a straightforward process once you establish a common denominator. In this problem, the expression changes from \( \frac{46 \sqrt{11}}{4} - \frac{\sqrt{11}}{4} \) to the simplified form.

• Ensure both fractions have the same denominator.
• Subtract the numerators and keep the denominator the same.

After that, you can simplify further if possible. Subtraction of fractions becomes more intuitive when each term shares a common base, as clearly demonstrated in this problem.