Problem 66 - College Algebra

Question

Simplify the radical expressions in Exercises \(61-68\) $$\sqrt[3]{12} \cdot \sqrt[3]{4}$$

Step-by-Step Solution

Answer

The simplified radical expression for \(\sqrt[3]{12} \cdot \sqrt[3]{4}\) is \(2 * \sqrt[3]{6}\).

Step 1: Prime Factorization

Break down the numbers under the cube roots into their prime factors. For \(12\), the prime factors are \(2, 2\), and \(3\). For \(4\), the prime factors are \(2\) and \(2\).

Step 2: Multiplication of Radicals

Multiply the numbers under the radical. That is, \(\sqrt[3]{12} * \sqrt[3]{4} = \sqrt[3]{12*4}\) which simplifies to \(\sqrt[3]{48}\).

Step 3: Simplify the Radical Expression

Now, it is necessary to simplify \(\sqrt[3]{48}\). The prime factors of \(48\) are \(2, 2, 2, 2, 3\). The cube root operation groups the prime factors into threes. Hence, one group of \(2, 2,\) and \(2\) can be taken out of the radical, leaving \(2 * \sqrt[3]{6}\) as the simplified radical expression.

Key Concepts

Prime Factorization

Understanding prime factorization is essential for working with radicals. It involves breaking down a number into its basic building blocks, which are prime numbers. Prime numbers are numbers greater than 1 that have no divisors other than 1 and themselves. For example, numbers like 2, 3, 5, 7 are prime numbers.

The purpose of finding the prime factors is to see the number in its simplest form, made entirely of primes. In the exercise, we need to break down the numbers 12 and 4 under the cube roots.

• For 12, the prime factors are: 2, 2, 3. (since 12 can be divided by 2 twice and then by 3)
• For 4, the prime factors are: 2, 2. (since 4 can be divided by 2 twice)

This step is crucial as it sets us up to efficiently simplify radicals by identifying how these factors can be reorganized when considering the cube roots.

Cube Roots

Cube roots provide us with a means to simplify expressions where a number is divided into three equal parts in terms of multiplication. In mathematical terms, \(\sqrt[3]{x}\) gives us a number, which when multiplied by itself three times, results in x.

When dealing with cube roots, like in this problem, we're looking to see how numbers can be "grouped" into threes. If any set of three identical primes exists under the cube root's radical, they can be "pulled out" of the radical.

Considering \(\sqrt[3]{48}\), once we have the prime factorization done, i.e., 2, 2, 2, 2, 3:

• The group here is a 2, 2, and 2, because three 2's multiplied give us part of the "ingredient" for pulling out of under the cube.

Thus, understanding cube roots means identifying these triple-groups. This helps a lot when simplifying radical expressions.

Simplifying Radicals

Simplifying radicals is all about making the expression as straightforward as possible. When you have a radical with a whole list of prime factors, you need to determine what "triple-groups" can be extracted if you're dealing with cube roots.

In our problem, we end up at \(\sqrt[3]{48}\). To simplify:

• From the earlier step, we know the prime factors of 48 are 2, 2, 2, 2, and 3.
• We find a group of three 2's.
• This group can be "taken out" of the radical. Once out, it becomes a simple integer multiplier outside of the cube root.

So, \(\sqrt[3]{48}\) simplifies to 2\(\sqrt[3]{6}\), because a group of three 2's forms the number 2 outside the radical, leaving 6 under the cube root. This process reduces the complexity of the expression you are dealing with by isolating components it’s easier to calculate with outside the cube root.