Problem 38 - Algebra 2 and Trigonometry

Question

In \(3-38\) , write each radical in simplest radical form. Variables in the radicand of an even index are non-negative. Variables occurring in the denominator of a fraction are non-zero. $$ \sqrt[4]{32 x^{5} y^{4}} $$

Step-by-Step Solution

Answer

The simplest radical form is \(2xy \sqrt[4]{2x}\).

Step 1: Identify the Radicand

The given radical is \( \sqrt[4]{32 x^{5} y^{4}} \). The radicand is the expression inside the root which is \(32 x^5 y^4\). We need to simplify this expression under a fourth root.

Step 2: Factor the Constant 32

The next step focuses on the numerical factor, 32. Notice that 32 can be simplified using its prime factorization: \( 32 = 2^5 \). This will be useful for simplifying the radical.

Step 3: Simplify Each Component of the Radicand

The goal is to express the radicand in a form where each factor is a perfect fourth power or close to it:- \( 32 = 2^5 = 2^4 \cdot 2 \),- \( x^5 = x^4 \cdot x \),- \( y^4 = (y^4) \).This allows us to separate components that can be taken out of the radical.

Step 4: Apply the Fourth Root to Each Factor

Use the property \( \sqrt[n]{a^n} = a \) for each fourth power factor:- \( \sqrt[4]{2^4} = 2 \ \sqrt[4]{2} \),- \( \sqrt[4]{x^4} = x \cdot \sqrt[4]{x} \),- \( \sqrt[4]{y^4} = y \). Therefore, \( \sqrt[4]{32 x^5 y^4} = 2xy \sqrt[4]{2x} \).

Step 5: Combine the Simplified Result

Combine the extracted factors and the remaining terms under the radical. The simplified form is: \( \sqrt[4]{32 x^5 y^4} = 2xy \sqrt[4]{2x} \).

Key Concepts

Simplified Radical Form

Simplifying radicals is about finding the simplest form of a given radical expression. This includes reducing the expression under the radical sign—also known as the radicand—so that it includes no perfect power of the radical's order. For a fourth root, as in our current problem, we're looking to express the radicand with any factors as fourth powers if possible.

• The aim is to simplify terms under the radical as much as possible, typically by factoring and using properties of roots.
• Once components are expressed as powers, it becomes straightforward to simplify the expression.
• After extracting these factors, we are left with an expression that can't be further simplified, providing us with the simplest radical form.

By doing so, the radical becomes easier to work with in mathematical calculations and expressions.

Factorization

Essential to simplifying radicals is the factorization process. Factorization involves breaking down numbers and variables into their smallest divisors. These divisors allow us to determine which portions of the radicand can be manipulated or removed when simplifying.

• For numbers, prime factorization is used. Numbers are expressed as products of prime numbers.
• For example, in the case of 32, it breaks down to \( 2^5 \).
• Variables are handled similarly by expressing them as powers, making it more convenient to work with roots later.

Factorization simplifies the process of solving complex radicals by isolating perfect power factors that can be easily extracted from under the radical sign.

Radicand

The radicand is the term inside a radical that we're looking to simplify. A clear understanding of the radicand is crucial since our main focus is on its simplification.

• In the problem \( \sqrt[4]{32 x^{5} y^{4}} \), the radicand is \( 32 x^5 y^4 \).
• Elements of the radicand are identified separately, including numerical factors and variable expressions.
• Each component must be analyzed to determine if it can be rewritten in a more manageable form for simplification.

By focusing on the radicand, we ensure that our calculations are accurate and lead to the simplest radical form possible.

Fourth Root

The fourth root is a specific type of radical where we're interested in extracting factors that are raised to powers divisible by four. This allows us to use one of the key properties of roots to remove terms from under the radical.

• The fourth root of a term \( a^4 \) is \( a \), which simplifies calculations.
• In our example, the fourth root \( \sqrt[4]{32 x^{5} y^{4}} \) is simplified by taking one of each fourth power factor out.
• Understanding how to work with fourth roots is essential whenever the radical index is four, making complex expressions easier to manipulate.

Recognizing and utilizing these powers ensures a streamlined path to simplifying expressions with higher-order radicals.