Problem 33 - Elementary and Intermediate Algebra

Question

Simplify each radical expression. All variables represent positive real numbers. $$ -\sqrt[5]{96 a^{4}} $$

Step-by-Step Solution

Answer

The simplified expression is \(-2 \cdot \sqrt[5]{3a^4}\).

Step 1: Prime Factorization

First, we'll find the prime factorization of 96. We have: \(96 = 2^5 \times 3\).

Step 2: Split the Radical Components

The expression inside the radical is \(96a^4 = 2^5 \times 3 \times a^4\). This can be broken down to \(-\sqrt[5]{2^5 \times 3 \times a^4}\).

Step 3: Simplify the Expression

Since we have \(2^5\) within the fifth root, we can simplify \(\sqrt[5]{2^5} = 2\). This simplifies our expression to \(-2\cdot\sqrt[5]{3a^4}\).

Step 4: Final Simplification

Since there are no other direct simplifications that can be performed within the fifth root, the simplified expression is \(-2 \cdot \sqrt[5]{3a^4}\).

Key Concepts

Prime Factorization

When simplifying radical expressions, understanding prime factorization is essential. Prime factorization involves breaking down a number into its basic building blocks, which are prime numbers. Prime numbers are integers greater than 1, with no divisors other than 1 and themselves. For example, the prime numbers include 2, 3, 5, 7, 11, and so forth.
The value 96 needs to be broken down using prime factorization:

• Start by dividing the number by the smallest prime, which is 2, because 96 is even.
• Continue dividing by 2 until no longer possible:
  • 96 ÷ 2 = 48
  • 48 ÷ 2 = 24
  • 24 ÷ 2 = 12
  • 12 ÷ 2 = 6
  • 6 ÷ 2 = 3

Finally, 3 is a prime number itself, so we end up with the prime factors for 96 as \(96 = 2^5 \times 3\).
This breakdown is crucial for simplifying radicals, especially when dealing with roots, such as fifth roots.

Fifth Roots

The fifth root of a number is a value that, when raised to the power of five, gives the original number. It is represented as \(\sqrt[5]{x}\). This concept extends the idea of square roots and cube roots to higher-level radicals.
For example, if you are given \(\sqrt[5]{32}\), you need to find a number which, when multiplied by itself five times, equals 32. Since \(2^5 = 32\), the fifth root of 32 is 2.
Similarly, in the exercise, you find the component \(\sqrt[5]{2^5}\). Because \(2^5 = 32\), simplifying \(\sqrt[5]{2^5}\) will give you 2 directly.

• This makes the process much more efficient.
• It helps in breaking down complex expressions into manageable parts.

Understanding fifth roots is crucial when simplifying expressions with higher-level radicals. It is a pivotal step to seamlessly transition into the simplification of complex radical expressions.

Simplification Steps

Simplifying radical expressions involves several key stages. Each stage is integral in achieving a clean and understandable result. The process requires a step-by-step breakdown:

• Begin by applying prime factorization to break down the components inside the radical.
• Second, analyze the expression within the root to identify complete power sets that match the type of root in question (e.g., fifth roots).
• In this example, we have \(\sqrt[5]{96a^4}\), simplified to \(\sqrt[5]{2^5 \times 3 \times a^4}\).
  • The component \(\sqrt[5]{2^5}\) simplifies to 2.
  • This reduces the expression to \(-2 \cdot \sqrt[5]{3a^4}\).

In the final simplified expression, \(-2 \cdot \sqrt[5]{3a^4}\), only \(3a^4\) remains under the fifth root, signifying the expression is as simple as it can be with basic algebraic steps.

• This systematic approach ensures that every part of the radical expression is broken down for simplification.
• Each stage is critical, making complex radical expressions easier to understand and work with.

By following these steps meticulously, you can confidently simplify even the most daunting of radical expressions.