Problem 78 - College Algebra Essentials

Question

In Exercises \(75-82,\) add or subtract terms whenever possible. $$3 \sqrt[3]{24}+\sqrt[3]{81}$$

Step-by-Step Solution

Answer

The simplified form of the expression \(3 \sqrt[3]{24} + \sqrt[3]{81}\) remains the same, as there are no like terms to combine, i.e, the answer is \(3 \sqrt[3]{24} + \sqrt[3]{81}\).

Step 1: Analyze the Cube Roots

Examine the cube roots in the expression: \(3 \sqrt[3]{24}\) and \(\sqrt[3]{81}\). These two cube roots have different radicands (24 and 81) and cannot be directly combined by addition or subtraction.

Step 2: Attempt to Simplify

Although you might try simplifying each individual cube root, neither 24 nor 81 can be simplified further while keeping both operations as cube roots.

Step 3: Conclusion

After those checks, it is concluded that the original expression cannot be simplified by combining the terms.

Key Concepts

Cube Roots

Cube roots refer to numbers that, when multiplied by themselves three times, yield the original number under the root. For instance, in the expression \(3\sqrt[3]{24}\), we need to find a number that gives 24 when cubed. Here, the number is the cube root of 24. Simplifying cube roots involves looking for perfect cube factors within the number. If a perfect cube is a factor of the number under the root, then you can express this cube root more simply.
To break it down:

• A cube root is not as straightforward as a square root since it involves multiplying a number by itself two more times.
• The cube root of a perfect cube, such as 8, is simple because 8 is 2 cubed (\(2^3 = 8\)), hence \(\sqrt[3]{8} = 2\).
• For numbers like 24 and 81, finding a perfect cube factor isn't possible, so they remain as they are.

In cases where the roots cannot be simplified, they remain in their original form. Recognizing when simplification isn't possible is an important algebraic skill.

Addition and Subtraction of Radicals

To add or subtract radicals, the radicand and the index should be identical. Think of it like adding similar objects: just as you can only combine apples with apples, the same goes for radicals. For example, to simplify expressions such as \(3\sqrt{2} + 5\sqrt{2}\), the radicals are the same so they can be combined, resulting in \(8\sqrt{2}\). However, for the expression \(3\sqrt[3]{24} + \sqrt[3]{81}\), the radicands are different (24 and 81).
Here's what you need to consider:

• Match the radicand and the radical index to directly add or subtract radicals.
• If the radicands or indices differ, like \(\sqrt{3}\) and \(\sqrt{5}\), they cannot be combined directly.
• Simplification of each term to see if further matching is possible is a good practice.

When radicals do not share the same roots or radicands, they remain separate, as was the case with our exercise example.

Radicands

Radicands are the numbers found inside the radical sign, the part of expressions like \(\sqrt[3]{24}\) and \(\sqrt[3]{81}\) that need evaluating. The radicand is fundamental because it determines what is being rooted. In this expression, the radicands 24 and 81 do not share common factors that are perfect cubes, meaning they can't be simplified further into a simpler radical with the same index.
Things to remember about radicands:

• The radicand affects whether the radicals can be added, subtracted, or simplified.
• If radicands can be broken down into simpler terms that share common factors, simplification is possible.
• Radicand similarity is key for adding and subtracting radicals efficiently.

An awareness of radicands allows you to plan your next algebraic steps, such as seeking common factors or considering other operations in expressions.