Problem 69 - Intermediate Algebra

Question

Simplify by combining like radicals. All variables represent positive real numbers. $$ 2 \sqrt[3]{16}-\sqrt[3]{54}-3 \sqrt[3]{128} $$

Step-by-Step Solution

Answer

The simplified expression is \(-11\sqrt[3]{2}\).

Step 1: Prime Factorize Each Radicand

First, we prime factorize each radicand under the cube roots to see if they can be simplified.\(16 = 2^4\),\(54 = 2 imes 3^3\),\(128 = 2^7\).

Step 2: Simplify Each Cube Root

Next, simplify each cube root using the prime factorization, considering that \(\sqrt[3]{x^3} = x\).For \(2 \sqrt[3]{16}\): \(\sqrt[3]{2^4} = 2 \cdot \sqrt[3]{2^3 \cdot 2} = 2 \cdot 2 \cdot \sqrt[3]{2} = 4\sqrt[3]{2}\).For \(-\sqrt[3]{54}\): \(-\sqrt[3]{2 \cdot 3^3} = -3\sqrt[3]{2}\).For \(-3 \sqrt[3]{128}\): \(-3\sqrt[3]{2^7} = -3 \cdot 2^2 \cdot \sqrt[3]{2} = -12\sqrt[3]{2}\).

Step 3: Combine Like Terms

Now, combine the simplified terms. All the terms have \(\sqrt[3]{2}\)\(4\sqrt[3]{2} - 3\sqrt[3]{2} - 12\sqrt[3]{2} = (4 - 3 - 12) \cdot \sqrt[3]{2} = -11\sqrt[3]{2}\)

Key Concepts

Prime Factorization

Prime factorization is a method used to express a number as a product of its prime numbers. Prime numbers are integers greater than 1 that have no divisors other than 1 and themselves. For example, the number 16 is expressed in terms of prime factors as \(16 = 2^4\), which means it is the product of four 2's. Similarly, 54 can be broken down as \(54 = 2 \times 3^3\), where both 2 and 3 are primes. This method helps us understand the building blocks of numbers, allowing us to simplify complex mathematical expressions, especially when dealing with roots.

Steps to find the prime factorization:

• Start with the smallest prime number (2) and divide the number until it can no longer be divided evenly by 2.
• Move to the next prime (3, 5, 7, etc.) and repeat the process.
• Continue dividing by primes until the result is a prime number.

This step is crucial for simplifying radicals since it allows us to pull out terms that are perfect cubes, squares, etc., thus making expressions easier to handle. In the exercise, prime factorization was used to simplify cube roots effectively.

Cube Roots

Cube roots are a form of radical expression represented by \(\sqrt[3]{x}\), where 3 signifies the root's degree. A cube root answers the question: "What number, when multiplied by itself three times, yields the original number?" For instance, the cube root of 8 is 2 because \(2^3 = 8\). Recognizing perfect cubes is essential for simplifying cube roots.

Simplifying cube roots involves:

• Using prime factorization to express the number under the root in terms of its prime factors.
• Identifying groups of three identical factors, since \(\sqrt[3]{x^3} = x\).
• Extracting these groups out of the radical, reducing the original expression.

Taking the example from the step-by-step solution, the cube root of 16 simplifies as \(\sqrt[3]{2^4}\). Recognizing that \(\sqrt[3]{2^3} = 2\), we simplify it to \(2 \cdot \sqrt[3]{2}\). This method helps to condense the expression by removing complete sets of cube factors, making it simpler to combine terms later on.

Combining Like Terms

Combining like terms is the process of adding or subtracting terms in an expression that have identical variables and exponents. This simplification technique is integral in algebra as it clarifies the expression and reveals its simplest form. When it comes to radicals, like terms must not only have the same variable but also the same radical part.

Here's how to combine like terms effectively:

• Identify all terms that have identical radical expressions.
• Add or subtract the coefficients of these like terms.
• Re-write the expression with the simplified coefficient and common radical.

In the exercise, all terms had \(\sqrt[3]{2}\) as the radical part, allowing us to focus on the coefficients: 4, -3, and -12. By adding these numerical values together: \(4 - 3 - 12\), we end up with \(-11\), leaving us with \(-11\sqrt[3]{2}\). This effectively condenses the expression into a single term with a clear coefficient, simplifying further operations or evaluations.