Problem 30 - Algebra and Trigonometry

Question

Simplify the expression. \(\sqrt[3]{24}-\sqrt[3]{81}\)

Step-by-Step Solution

Answer

The simplified expression is \(-\sqrt[3]{3}\).

Step 1: Identify and Simplify Cube Roots

First, we identify the cube roots of the numbers inside the radicals. Since 24 and 81 are not perfect cubes, we seek their prime factorization to simplify: \(24 = 2^3 \times 3\) and \(81 = 3^4\). Thus the expression transforms as: \(\sqrt[3]{24} = \sqrt[3]{2^3 \times 3} = 2\sqrt[3]{3}\) and \(\sqrt[3]{81} = \sqrt[3]{3^4} = 3\sqrt[3]{3}\).

Step 2: Substitute and Simplify Expression

Substitute the simplified cube roots back into the expression: \(2\sqrt[3]{3} - 3\sqrt[3]{3}\).

Step 3: Factor Out the Common Term

Observe that \(\sqrt[3]{3}\) is a common factor in both terms. Factor it out: \((2 - 3)\sqrt[3]{3}\).

Step 4: Simplify the Expression

Calculate \(2 - 3 = -1\). Substitute this into the expression to get \(-1\sqrt[3]{3}\), which simplifies to \(-\sqrt[3]{3}\).

Key Concepts

Cube Roots

Understanding cube roots is pivotal in simplifying expressions that involve cubic radicals. A cube root, denoted as \(\sqrt[3]{x}\), is a number that, when multiplied by itself three times, gives \(x\) back. Unlike square roots, where we need pairs to form a complete square, for cube roots, we're looking for triplets. This is a little more complex because we're working with a set of three factors of the same number.

To simplify a cube root expression, it's crucial to recognize whether the number inside the radical is a perfect cube or not. Perfect cubes have whole numbers (like 8, 27, 64) as their cube roots. However, numbers like 24 and 81, while not perfect cubes, can still be dealt with by breaking them down—this is where prime factorization becomes beneficial. Once expressed in its prime components, you can often simplify by grouping factors into triples, which essentially "pulls out" a cube root.

Prime Factorization

Prime factorization involves expressing a number as a product of its prime numbers. This method is instrumental when dealing with cube roots of non-perfect cubes, such as 24 and 81 in our problem.

To do this, split down the number into its basic building blocks—primes. For example:

• For 24, breaking it down gives us: \(24 = 2 \times 12 = 2 \times 2 \times 6 = 2 \times 2 \times 2 \times 3 = 2^3 \times 3\).
• For 81, it looks like this: \(81 = 3 \times 27 = 3 \times 3 \times 9 = 3 \times 3 \times 3 \times 3 = 3^4\).

Prime factorization makes it easier to identify patterns or factors that can repeatedly fit within the cube, allowing us to simplify further. Knowing the prime factorization gives us a roadmap to pulling out parts of the solution efficiently and effectively.

Common Factors

When you've managed to break down terms inside cube roots and reduce them as far as possible, the next step is identifying common factors. A common factor occurs in an expression when the same term exists in multiple parts.

In our expression, after simplifying, both terms ended up having \(\sqrt[3]{3}\) as a common factor, specifically expressed as \(2\sqrt[3]{3}\) and \(3\sqrt[3]{3}\). You can factor it out by combining it with the other coefficients of the expression:

• This results in factoring out \(\sqrt[3]{3}\) from \(2\sqrt[3]{3} - 3\sqrt[3]{3}\).
• After pulling out \(\sqrt[3]{3}\), you're simply left with \((2 - 3)\sqrt[3]{3}\).

Recognizing and factoring out common terms allows the expression to be written in a much simpler form, often leading to more easier arithmetic operations thereafter.