Problem 6 - Algebra for College Students

Question

Use the distributive property to help simplify each of the following. \(-2 \sqrt{20}-7 \sqrt{45}\)

Step-by-Step Solution

Answer

-25\sqrt{5}

Step 1: Factor the Radicals

First, we need to simplify the square roots by factoring them into their prime components.- \( \sqrt{20} = \sqrt{4 \times 5} = \sqrt{4} \times \sqrt{5} = 2\sqrt{5} \)- \( \sqrt{45} = \sqrt{9 \times 5} = \sqrt{9} \times \sqrt{5} = 3\sqrt{5} \)

Step 2: Apply the Distributive Property

Next, substitute the simplified radicals back into the original expression: \(-2(2\sqrt{5}) - 7(3\sqrt{5})\).Apply the distributive property to distribute the coefficients:- \(-2 \times 2\sqrt{5} = -4\sqrt{5}\)- \(-7 \times 3\sqrt{5} = -21\sqrt{5}\)

Step 3: Combine Like Terms

Now, combine the terms that have the same radical by adding their coefficients:\(-4\sqrt{5} - 21\sqrt{5} = (-4 - 21)\sqrt{5} = -25\sqrt{5}\).

Key Concepts

Simplifying Radicals

When working with radicals, it's often helpful to simplify them by breaking them down into their prime components. This is because a simplified radical is easier to work with in equations and expressions. Let's consider the square root of 20, which we can factor into 4 and 5. Since \(4 = 2^2\), this allows us to simplify \(\sqrt{20}\) to \(2\sqrt{5}\). Similarly, for \(\sqrt{45}\), we factor it into 9 and 5. Since \(9 = 3^2\), we can rewrite \(\sqrt{45}\) as \(3\sqrt{5}\).

• Look for perfect squares in the number under the radical.
• Factor these out to simplify the radical further.
• This simplification helps set up expressions for easier manipulation in subsequent steps, like applying the distributive property or combining like terms.

Prime Factorization

Prime factorization is the process of breaking down a number into its basic building blocks—prime numbers. Understanding this is crucial for simplifying radicals, as demonstrated in expressions like \(\sqrt{20}\) and \(\sqrt{45}\) from the exercise. When you factor a number, start by dividing it by the smallest prime number, 2, and continue with the next primes (3, 5, 7, etc.) until you've expressed it entirely as a product of primes.

• \(20 = 2 \times 2 \times 5\)
• \(45 = 3 \times 3 \times 5\)

Use these factorizations to simplify radicals by pulling out squares of these primes. This highlights the importance of recognizing prime factors as they lead directly to simplified forms in mathematical expressions.

Combining Like Terms

Once the radicals are simplified, the next step in our process involves combining like terms. In algebra, like terms are terms whose variables (and corresponding exponents) are the same. Here, both terms \(-4\sqrt{5}\) and \(-21\sqrt{5}\) are like because they contain the same radical part \(\sqrt{5}\).

• Identify terms with the same radical or variable.
• Add or subtract the coefficients of these terms while keeping the radical part unchanged.
• In this case, \(-4\sqrt{5} - 21\sqrt{5} = -25\sqrt{5}\)

This simplification step is crucial for finalizing expressions and making them more compact and easier to understand.