Problem 42 - Intermediate Algebra

Question

Simplify each radical expression. All variables represent positive real numbers. $$ \sqrt[4]{\frac{5 y^{12}}{64}} $$

Step-by-Step Solution

Answer

The expression simplifies to \( \frac{y^3 \cdot \sqrt[4]{5}}{2} \).

Step 1: Identify Components

First, we identify the components within the fourth root. The expression is \( \sqrt[4]{\frac{5 y^{12}}{64}} \). We have the numerator \( 5 y^{12} \) and the denominator \( 64 \).

Step 2: Simplify the Denominator

Since 64 is a perfect fourth power, we simplify it. We recognize that \( 64 = 2^6 \), and thus \( \sqrt[4]{64} = 2^{6/4} = 2^{3/2} = 2 \).

Step 3: Simplify the Numerator

For the numerator \(5 y^{12}\), we separate the constant 5 and the variable term \(y^{12}\). Since \( y^{12} \) is a perfect fourth power, recognize \( y^{12} = (y^3)^4 \). Therefore, \( \sqrt[4]{y^{12}} = y^3 \).

Step 4: Combine Results

Combine the simplified results: the fourth root of the numerator is \( y^3 \), and the fourth root of the denominator is \( 2 \). Therefore, the expression simplifies to \( \frac{y^3 \cdot \sqrt[4]{5}}{2} \).

Step 5: Express in Final Form

We write the simplified expression in its final form. Since the constant 5 in the numerator does not simplify further under the fourth root, the final simplification is \( \frac{y^3 \cdot \sqrt[4]{5}}{2} \).

Key Concepts

Simplifying Radicals

In mathematics, radical expressions often seem daunting, but with a few simple steps, they become manageable. A radical expression involves roots such as square roots, cube roots, or fourth roots. Simplifying radicals means expressing a radical in its simplest form. The key is to break down the numbers or expressions under the root to identify and simplify perfect powers. When simplifying, remember:

• Look for terms that are perfect squares, cubes, etc., depending on the radical type.
• Redistribute the terms to simplify using properties of exponents.
• If the root is a fourth root, break it into components that make use of the rule \( \sqrt[n]{a} = a^{1/n} \).

Thus, understanding whether parts of the expression can be expressed as powers of the radical's index, say four for fourth roots, helps in reducing complexity. Take, for instance, simplifying \( \sqrt[4]{64} \). Recognizing 64 as \( 2^6 \), you realize \( \sqrt[4]{64} \), is \( 2^{3/2} = 2 \). So, breaking down numbers into their prime factors, and expressing them as powers helps in elegant simplification of radicals.

Fourth Roots

Fourth roots require understanding of radical principles and are particularly useful in algebra where they simplify expressions involving variables raised to high powers. To deal with fourth roots:

• Look for numbers or expressions that can be decomposed into powers of four.
• Utilize the property \( x^{m/n} = (x^m)^{1/n} \), which applies neatly when you want to separate a term from under the radical.
• Apply this by simplifying known figures and algebraic terms. For instance, the fourth root of a perfect fourth power, like \( y^{12} \) which is \( (y^3)^4 \), simplifies directly to \( y^3 \).

Under this method, the radical is removed step by step, thereby ensuring that simpler terms emerge and the expression becomes easier to manage. This primary breakdown of numbers into their constituent factors underpins all successful simplifications.

Variable Exponents

Dealing with variable exponents is a key skill, especially when simplifying expressions inside radicals. Variable exponents frequently appear in algebra, and utilizing power rules is crucial for simplification.Let's look into how to manage them effectively:

• Recognize their structure so they can be simplified using laws of exponents.
• Remember that a variable exponent such as \(y^{12}\) treated under a fourth root \(\sqrt[4]{y^{12}}\), hinges on factoring it as a power of four: \((y^3)^4\).
• This turns into a straightforward simplification as \( y^3 \) since \( \sqrt[4]{y^{12}} = \sqrt[4]{(y^3)^4}= y^3 \). Use this approach consistently for simplifying expressions with variable exponents.

By breaking down and managing exponents in such a manner, equations become manageable and provide clarity to what initially seemed complex. Understanding the properties of exponents is thus invaluable in algebra and beyond.