Problem 72 - College Algebra

Question

In Exercises \(69-76,\) add or subtract terms whenever possible. $$3 \sqrt[3]{24}+\sqrt[3]{81}$$

Step-by-Step Solution

Answer

So, \(3 \sqrt[3]{24}+\sqrt[3]{81} = 9.

Step 1: Simplify The Cube Roots

The first step is to simplify the cube roots of the terms. Any perfect cube number with a base that is divisible by the base of our given value will be simplified. For instance, \(24=2^3*3\), thus we have a cube root of a perfect cube part, which is \(2\). Same applies to \(81=3^4\), for the cube root we take out the base \(3\) once, and we get remainder \(3^1\). Therefore, the equation simplifies to \(3 \sqrt[3]{2^3*3}+\sqrt[3]{3^3*3}\)The equation further simplifies to \(3 * 2 + 3\) = \(6+3\)

Step 2: Addition of The Terms

The second step is to combine the terms we obtained after simplification. So, \(6+3 = 9\).

Key Concepts

Perfect Cube Numbers

Perfect cube numbers are numbers that result from multiplying a number by itself twice (i.e., raising it to the power of three). They are the three-dimensional counterparts to perfect squares. Some examples include:

• 1 because: \(1 \times 1 \times 1 = 1\)
• 8 because: \(2 \times 2 \times 2 = 8\)
• 27 because: \(3 \times 3 \times 3 = 27\)
• 64 because: \(4 \times 4 \times 4 = 64\)

Finding perfect cubes within numbers can simplify their cube roots. In the example with cube roots, we found that part of the number 24 (\(2^3\)) and most of 81 (\(3^3\)) are perfect cubes. By recognizing these, simplification becomes more straightforward. Identify the largest perfect cube factor when dealing with cube roots to make calculations easier.

Addition and Subtraction of Radicals

Radicals are numbers expressed with a root symbol. When performing operations like addition and subtraction with them, it's crucial for radicals to have the same index and radicand (the number under the root). If they're not identical, you must simplify them first or the operation can't be performed.

Just like with regular fractions, you can only directly add or subtract similar radicals. After simplifying the cube roots in your expression, you'll see if the terms have the same radicand, allowing you to add them directly.

Our initial problem, \(3 \sqrt[3]{24} \text{ and } \sqrt[3]{81}\), didn't have a common radicand. After simplification, however, both terms were reduced to constants, enabling straightforward addition: \(6 + 3\). The key is simplifying such that the radicals are compatible for basic arithmetic operations.

Cube Root Simplification

Simplifying cube roots is largely about finding the prime factors of the number under the root and then grouping them in threes. The idea is similar to simplifying square roots, but in this case, you identify factors that can be grouped into sets of three.

Let's take \(\sqrt[3]{24}\) as an example. First, break down 24 into its prime factors: \(2^3 \times 3\). The grouping here involves the three 2's, which allows you to take a 2 out of the cube root, simplifying it to \(2\sqrt[3]{3}\).

• Next, consider \(\sqrt[3]{81}\). The factorization of 81 is \(3^4\), where you can take three 3's out, leaving you with \(3\sqrt[3]{3}\).

Simplifying cube roots by extracting perfect cube factors, simplifies the arithmetic. As we've seen, both terms were reduced to single numbers that could easily be added or subtracted. This process is crucial for managing complex expressions with cube roots.