Problem 82 - Algebra A Combined Function

Question

$$ \sqrt[4]{\frac{x^{4}}{16}} $$$$ \sqrt[4]{\frac{y^{4}}{81 x^{4}}} $$

Step-by-Step Solution

Answer

The simplified expressions are \( \frac{x}{2} \) and \( \frac{y}{3x} \).

Step 1: Simplify the First Expression

The first expression is \( \sqrt[4]{\frac{x^4}{16}} \). By using the property \( \sqrt[n]{\frac{a}{b}} = \frac{\sqrt[n]{a}}{\sqrt[n]{b}} \), we can separate this into two separate fourth roots: \( \frac{\sqrt[4]{x^4}}{\sqrt[4]{16}} \).

Step 2: Evaluate Individual Fourth Roots for the First Expression

Now evaluate each part: \( \sqrt[4]{x^4} = x \) because taking the fourth root and raising to the fourth power cancels out. For \( \sqrt[4]{16} = 2 \) since \( 2^4 = 16 \). Thus, \( \frac{\sqrt[4]{x^4}}{\sqrt[4]{16}} = \frac{x}{2} \).

Step 3: Simplify the Second Expression

The second expression is \( \sqrt[4]{\frac{y^4}{81x^4}} \). Using the same property, we have \( \frac{\sqrt[4]{y^4}}{\sqrt[4]{81x^4}} \).

Step 4: Evaluate Individual Fourth Roots for the Second Expression

For \( \sqrt[4]{y^4} = y \) and \( \sqrt[4]{81x^4} \), separately calculate \( \sqrt[4]{81} = 3 \) since \( 3^4 = 81 \), and \( \sqrt[4]{x^4} = x \). Thus, the expression becomes \( \frac{y}{3x} \).

Key Concepts

Fractional Exponents

When dealing with radicals such as fourth roots, using fractional exponents can simplify expressions and calculations. A fractional exponent represents both a power and a root. For example, \( \sqrt[4]{x^4} \) can be written as \( x^{4/4} \), which simplifies directly to \( x^1 \) or just \( x \). This relationship stems from the rule that \( a^{m/n} \) is the same as \( \sqrt[n]{a^m} \).
This means that when you see an expression like \( \sqrt[4]{16} \), it can instead be written as \( 16^{1/4} \). Since \( 16 \) can be broken down into \( 2^4 \), \( 16^{1/4} \) simplifies to 2.
So, understanding that fractional exponents are another way to view radicals helps with algebraic manipulation and solving processes.

Simplification Techniques

Simplification plays a crucial role when solving radical expressions. It involves reducing expressions to their most basic form, which can be vital for clearer, easier calculations.
The core idea consists of breaking down the expression components to their elementary forms. Using the property \( \sqrt[n]{\frac{a}{b}} = \frac{\sqrt[n]{a}}{\sqrt[n]{b}} \), any complicated expression can be consistently simplified by examining the numerator and the denominator individually.

• For example, breaking down \( \sqrt[4]{\frac{x^4}{16}} \) into \( \frac{\sqrt[4]{x^4}}{\sqrt[4]{16}} \) allows us to deal with \( x^4 \) and 16 separately.
• Similarly, \( \sqrt[4]{x^4} = x \) because the root and power cancel each other out.
• Applying this to \( \sqrt[4]{16} \), where 16 is \( 2^4 \), results directly in 2.

This process results in the simplified expression \( \frac{x}{2} \). Clearly separating components is not only a step-by-step practice but also a mindset that helps avoid mistakes.

Root Manipulation

Root manipulation involves operations and transformations with radical expressions to achieve a simplification or a clearer view. This technique is fundamental when handling complex fractions under roots.
To illustrate, let's consider the expression \( \sqrt[4]{\frac{y^4}{81x^4}} \). By rewriting it as \( \frac{\sqrt[4]{y^4}}{\sqrt[4]{81x^4}} \), manipulation becomes simpler and more organized.
Within the expression:

• Handling \( y^4 \) and \( x^4 \) within fourth roots becomes straightforward, transforming them into \( y \) and \( x \) respectively, as the roots cancel out.
• For the number 81, known to be \( 3^4 \), its fourth root simplifies to 3.
• Therefore, the expression turns into \( \frac{y}{3x} \), showcasing straightforward, logical root manipulations.

Recognizing the means to reorganize and rewrite radical expressions is a key element, enabling both ease in solving and understanding the structure of mathematical problems.