Problem 16 - Algebra for College Students

Question

Use the distributive property to help simplify each of the following. \(\frac{-2 \sqrt{20}}{3}+\frac{3 \sqrt{45}}{4}-\frac{5 \sqrt{80}}{6}\)

Step-by-Step Solution

Answer

The simplified expression is \(\frac{-29\sqrt{5}}{12}\).

Step 1: Simplify Each Square Root

Start by simplifying the square roots in each term. The term \(-2\sqrt{20}\) can be broken down as \(-2\sqrt{4 \times 5} = -2\times 2\sqrt{5} = -4\sqrt{5}\). Similarly, \(3\sqrt{45}\) becomes \(3\sqrt{9 \times 5} = 3\times 3\sqrt{5} = 9\sqrt{5}\). Lastly, \(5\sqrt{80}\) simplifies to \(5\sqrt{16 \times 5} = 5 \times 4\sqrt{5} = 20\sqrt{5}\).

Step 2: Simplify the Fraction Coefficients

Substitute the simplified square roots back into the fractions. This gives us \(\frac{-4\sqrt{5}}{3}\), \(\frac{9\sqrt{5}}{4}\), and \(\frac{20\sqrt{5}}{6}\).

Step 3: Find a Common Denominator

Identify a common denominator for the fractions \(\frac{-4\sqrt{5}}{3}\), \(\frac{9\sqrt{5}}{4}\), and \(\frac{20\sqrt{5}}{6}\). The least common multiple of 3, 4, and 6 is 12.

Step 4: Convert Each Fraction to Have the Common Denominator

Convert each fraction to have a denominator of 12. For \(\frac{-4\sqrt{5}}{3}\), multiply both the numerator and denominator by 4 to get \(\frac{-16\sqrt{5}}{12}\). For \(\frac{9\sqrt{5}}{4}\), multiply both the numerator and denominator by 3 to get \(\frac{27\sqrt{5}}{12}\). For \(\frac{20\sqrt{5}}{6}\), multiply both the numerator and denominator by 2 to get \(\frac{40\sqrt{5}}{12}\).

Step 5: Combine the Fractions

Now that all terms have the same denominator, add them together: \(\frac{-16\sqrt{5} + 27\sqrt{5} - 40\sqrt{5}}{12}\).

Step 6: Simplify the Numerator

Simplify the expression in the numerator: \(-16\sqrt{5} + 27\sqrt{5} - 40\sqrt{5} = -29\sqrt{5}\). Thus, the fraction becomes \(\frac{-29\sqrt{5}}{12}\).

Step 7: Final Answer

The simplified expression using the distributive property is \(\frac{-29\sqrt{5}}{12}\).

Key Concepts

simplifying square roots

Simplifying square roots involves breaking down the expression under the square root to its prime factors, which makes it much easier to work with. For example, consider the square root of 20, represented as \( \sqrt{20} \). To simplify \( \sqrt{20} \), we first break down 20 into its factors: \( 4 \times 5 \). Since 4 is a perfect square, we can take its square root, resulting in \( 2\sqrt{5} \).
Similarly, for \( \sqrt{45} \), we notice that 45 can be rewritten as \( 9 \times 5 \). Because 9 is a perfect square (\( 3^2 \)), we simplify \( \sqrt{45} \) to \( 3\sqrt{5} \). For \( \sqrt{80} \), we factor it as \( 16 \times 5 \), where 16 is \( 4^2 \), so \( \sqrt{80} \) simplifies to \( 4\sqrt{5} \).
By simplifying these square roots, we make the mathematical expressions easier to combine and solve in subsequent steps. This simplification process is crucial for reducing complex expressions involving square roots to their simplest form.

common denominator

Finding a common denominator is essential when adding or subtracting fractions. This involves finding the least common multiple (LCM) of the denominators involved.
In our case, we have three fractions: \( \frac{-4\sqrt{5}}{3} \), \( \frac{9\sqrt{5}}{4} \), and \( \frac{20\sqrt{5}}{6} \). To add these fractions together, we need to determine a number that is a multiple of all three denominators: 3, 4, and 6. The LCM of these numbers is 12.
Once we have our common denominator, each fraction is converted to have this denominator. For instance, multiply both numerator and denominator of \( \frac{-4\sqrt{5}}{3} \) by 4 to get \( \frac{-16\sqrt{5}}{12} \).
This process of converting fractions ensures that we can easily add or subtract them by aligning their denominators.

fraction simplification

After aligning the fractions with a common denominator, the next step is to simplify. This process involves combining the numerators over the common denominator.
In the example, once each fraction has a common denominator of 12, we have: \( \frac{-16\sqrt{5}}{12} \), \( \frac{27\sqrt{5}}{12} \), and \( \frac{40\sqrt{5}}{12} \). We then combine these by adding or subtracting their numerators: \(-16\sqrt{5} + 27\sqrt{5} - 40\sqrt{5} = -29\sqrt{5} \).
Finally, we write the resulting expression as \( \frac{-29\sqrt{5}}{12} \).
Fraction simplification helps us find the most reduced form of expressions, eliminating unnecessary complexity and making further calculations more manageable.