Problem 31 - Algebra for College Students

Question

Change each radical to simplest radical form. \(-6 \sqrt{20}\)

Step-by-Step Solution

Answer

The simplest radical form is \(-12 \sqrt{5}\).

Step 1: Identify the Radicand

Examine the radical expression \(-6 \sqrt{20}\). The number under the square root symbol is called the radicand. In this case, the radicand is 20.

Step 2: Factor the Radicand into Prime Factors

Find the prime factors of the radicand 20. The prime factorization of 20 is \(20 = 2 \times 2 \times 5 = 2^2 \times 5\).

Step 3: Simplify the Radicand

Express \(\sqrt{20}\) using the prime factors found in step 2: \(\sqrt{20} = \sqrt{2^2 \times 5}\). Use the property of square roots that states \(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\) to separate the square root into two parts: \(\sqrt{2^2} \times \sqrt{5}\).

Step 4: Simplify the Expression

Since \(\sqrt{2^2} = 2\), replace this in the expression: \(\sqrt{20} = 2 \times \sqrt{5}\). The original expression \(-6 \sqrt{20}\) can then be rewritten as \(-6 \times 2 \times \sqrt{5}\).

Step 5: Calculate the Simplified Radical Form

Multiply \(-6\) by \(2\) to simplify the expression: \(-6 \times 2 = -12\). Thus, \(-6 \sqrt{20} = -12 \sqrt{5}\).

Key Concepts

Radicand

In a radical expression like \(-6 \sqrt{20}\), the radicand is the number situated under the square root symbol. In this case, our radicand is 20. The radicand gives the expression its 'root' value, so understanding what you are working with is crucial. Identifying the radicand is the first step in simplifying radical expressions. This lets you know the starting point of your calculations. If you start without identifying the radicand, you might find it challenging to proceed through the steps accurately.

Prime Factorization

Prime factorization involves breaking down a number into its most basic building blocks - the prime numbers. For the radicand 20, prime factorization involves rewriting it as a product of prime numbers. Here's how we do it:

• First, note that 20 is an even number, which means it's divisible by 2. Divide by 2 to get 10.
• 10 is also even, so divide by 2 again to get 5.
• 5 is already a prime number, so the factorization process stops here.

Thus, the prime factorization of 20 is expressed as \(20 = 2 \times 2 \times 5 = 2^2 \times 5\). Prime factorization helps in breaking down complex radicals into simpler, manageable parts.

Property of Square Roots

The property of square roots is a handy mathematical principle for simplifying radicals. It states that \(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\). This is essential when you have a radicand with factors you want to simplify separately.Using our example \(-6 \sqrt{20}\), we can use this property to break down \(\sqrt{20}\). From the prime factorization, we know 20 = \(2^2 \times 5\).Apply property: \(\sqrt{20}\) becomes \(\sqrt{2^2} \times \sqrt{5}\).Since \(\sqrt{2^2} = 2\), it simplifies to \(2 \times \sqrt{5}\). This property significantly eases the task of simplifying radical expressions.

Simplifying Expressions

Simplifying expressions means reducing them to their simplest form. This often involves removing square roots from the denominator or combining terms to make calculations easier.Using the example \(-6 \sqrt{20}\), after applying the property of square roots and prime factorization, you get \(-6 \times 2 \times \sqrt{5}\). By multiplying the constants, \(-6 \,and\,\ 2\), you get \(-12\).Thus, the expression simplifies to \(-12 \sqrt{5}\). Simplifying makes it more straightforward to work with an expression. It helps you to concentrate on the most pertinent parts of the problem, which are the numerical coefficients and the leftover radicals.