Problem 33 - 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 \(-2a^{4/5} \times \sqrt[5]{3}\).

Step 1: Prime Factorization of 96

First, identify the prime factorization of 96. Notice that 96 can be divided by 2, which is the smallest prime number, repeatedly until it reaches 1: \[ 96 = 2^5 imes 3 \]

Step 2: Express the Radical Expression

Rewrite the given radical expression \(-\sqrt[5]{96 a^4}\) using the prime factorization: \[ -\sqrt[5]{(2^5 \times 3) a^4} \]

Step 3: Simplify the Radical

Recognize that each factor raised to the fifth power under a fifth root can be separated: \[ -\sqrt[5]{2^5} \times \sqrt[5]{3} \times \sqrt[5]{a^4} \]Since \(\sqrt[5]{2^5} = 2\), simplify as follows: \[ -2 \times \sqrt[5]{3} \times \sqrt[5]{a^4} \]

Step 4: Split the Variables

Express \(\sqrt[5]{a^4}\) using exponents: \[ \sqrt[5]{a^4} = a^{4/5} \]

Step 5: Combine and Simplify

Combine all simplified parts: \[ -2a^{4/5} \times \sqrt[5]{3} \]This is the simplified version of the original radical expression.

Key Concepts

Prime Factorization

Prime factorization is breaking down a whole number into its basic building blocks called prime numbers. A prime number is a number that is greater than 1 and cannot be formed by multiplying two smaller natural numbers. To get the prime factorization of a number, you divide it by the smallest prime until you can't divide anymore, except by 1.

For example, consider the number 96. We start dividing by 2, the smallest prime:

• Divide 96 by 2 to get 48
• Divide 48 by 2 to get 24
• Divide 24 by 2 to get 12
• Divide 12 by 2 to get 6
• Divide 6 by 2 to get 3

Now, 3 is a prime number, and can't be divided by 2, so we're done. The prime factorization of 96 is thus expressed as: \[ 96 = 2^5 \times 3 \]Understanding how to find the prime factorization is a key step in simplifying radicals as it allows us to identify and break down numbers into factors, making the simplification process possible.

Simplifying Radicals

Simplifying radicals involves breaking down the expression under the root into its simplest form. This often requires the use of prime factorization. Once we have the factors, we can often "take out" full sets of factors outside of the radical. In simpler terms, you're trying to pull out any complete sets and reduce what's inside the root symbol.

When dealing with a fifth root, like \(-\sqrt[5]{96 a^4}\), the goal is to identify factors that match the root being taken: groups of five identical numbers or variables. Starting with \(\sqrt[5]{2^5}\), the 2 inside can be entirely removed from under the root because \(2^5\) equals exactly what's needed for a fifth power to leave the radical.

For the expression \(-\sqrt[5]{96 a^4}\), the process looked like:

• Identifying \(\sqrt[5]{2^5}\) and simplifying it to 2.
• Handling \(\sqrt[5]{a^4}\) since it doesn't reach the power of 5 needed for complete simplification, leaving it as \(a^{4/5}\).
• Combining these simplifications to get \(-2a^{4/5}\times\sqrt[5]{3}\).

Being able to factor correctly and recognize these complete sets allows for the root to "release" a number or variable, effectively simplifying the expression.

Fifth Roots

Fifth roots involve finding a number that, when raised to the power of 5, gives the original number or expression inside the root. The goal when simplifying a fifth root, like any radical, is to reduce what's inside by finding powers of 5.

A number in the form of \(\sqrt[5]{x}\) asks, "What number times itself five times equals x?"

In practical terms, when simplifying, say \(\sqrt[5]{a^4}\), as found in the step-by-step solution, the exponent is close to 5 (but not quite there), so it's expressed as \(a^{4/5}\). If you had something like \(\sqrt[5]{a^5}\), it would simplify directly to \(a\) because 5 is a power that matches the fifth root.

The key takeaway here is recognizing when the internal components of the root expression reach a multiple of 5, facilitating complete or partial extraction out of the root. It opens up a clearer path to simplify complex radical expressions like \(-\sqrt[5]{96 a^4}\) by applying these simplification rules systematically.