Problem 34 - College Algebra

Question

Simplify each expression. $$3 \sqrt[3]{-432}+\sqrt[3]{16}$$

Step-by-Step Solution

Answer

The simplified expression is \(-18 + 2 \cdot \sqrt[3]{2}\).

Step 1: Identify the cube roots

First, recognize that you need to find the cube roots of the numbers inside the radical expressions: \ \(\sqrt[3]{-432}\) and \(\sqrt[3]{16}\).

Step 2: Simplify the cube root of -432

Break down -432 into its prime factors: \(-432 = -2^4 \times 3^3\). The cube root of \(-432\) is \ \(\sqrt[3]{-2^4 \times 3^3} = \sqrt[3]{3^3} \cdot \sqrt[3]{-2^4} = -3 \cdot 2 = -6\).

Step 3: Simplify the cube root of 16

Factor 16 as \(2^4\), so the cube root is \ \(\sqrt[3]{16} = \sqrt[3]{2^4} = 2^{4/3} = 2^{1+1/3} = 2 \cdot \sqrt[3]{2}\).

Step 4: Combine the simplified expressions

Now substitute back the simplified cube roots into the original expression: \ \(3 \times (-6) + 2 \cdot \sqrt[3]{2}\). This gives \ \(-18 + 2 \cdot \sqrt[3]{2}\).

Key Concepts

Cube roots

In mathematics, a cube root of a number is a value that, when multiplied by itself twice, results in the original number. For example, the cube root of 8 is 2, because \[2 \times 2 \times 2 = 8\]Cube roots are denoted by the symbol \( \sqrt[3]{} \). To find the cube root, you can use numerical estimation, factorization, or technology like calculators for more complex numbers.
Finding the cube root involves understanding how multiplication can be reversed into a root, making it a fundamental aspect of simplifying mathematical expressions. In our original problem, we needed to find the cube roots of \(-432\) and \(16\). Factoring helps us simplify these cube roots and solve the problem more efficiently.

Radical expressions

Radical expressions are mathematical expressions that include a radical sign \( \sqrt{} \). Examples include square roots \( \sqrt{9} \), cube roots \( \sqrt[3]{27} \), and higher roots. They allow us to work with roots of numbers that cannot be expressed as simple fractions.Understanding how to simplify radical expressions involves:

• Identifying the radical sign and the number it affects
• Using factorization to make simplification possible
• Combining like terms where applicable

Radical expressions can be intuitive when broken down into simpler parts and matched with their corresponding roots, such as turning \(16\) into \(2^4\) to find its cube root. This approach gives a clear path to solve more complicated problems.

Prime factorization

Prime factorization breaks down a number into its prime number components. Primes are numbers that are only divisible by 1 and themselves, such as 2, 3, 5, etc.To factor a number like \(-432\), we would:

• Find the prime factors:
  \( -432 = -2^4 \times 3^3 \)
• Apply these to simplify cube roots or other expressions
  

Prime factorization is crucial for simplifying expressions because it reveals the smallest building blocks of numbers, facilitating operations like finding cube roots by exposing underlying factors.
Through such factorization, cube root calculations become streamlined, reducing complexity by focusing directly on manageable components.

Mathematical operations

Understanding and performing mathematical operations is key in manipulating numbers and expressions effectively. Common operations include addition, subtraction, multiplication, and division, which build arithmetic foundations.In our exercise, operations help us combine the results of cube roots:

• Multiplying simplified cube root results: \(-3\) multiplied by \(2\) gives \(-6\)
• Adding or subtracting terms: here, \(-18 + 2 \cdot \sqrt[3]{2}\)

This mathematical dexterity allows us to solve expressions efficiently by focusing on steps that bring us to a clearer, simplified result.
Working through these operations in sequence ensures accuracy and comprehensibility, making even complex problems manageable with practice.