Problem 39 - Intermediate Algebra

Question

Simplify each radical expression. All variables represent positive real numbers. $$ \sqrt[4]{\frac{3}{625}} $$

Step-by-Step Solution

Answer

\( \frac{\sqrt[4]{3}}{5} \)

Step 1: Recognize the Radical Expression

We start by identifying the radical expression. We have \( \sqrt[4]{\frac{3}{625}} \), which is a fourth root of the fraction \( \frac{3}{625} \).

Step 2: Simplify the Denominator

Notice that 625 is a perfect fourth power since \( 625 = 5^4 \). Therefore, we can simplify \( \sqrt[4]{625} \) to \( 5 \).

Step 3: Simplify the Radical Expression

The expression becomes \( \sqrt[4]{\frac{3}{625}} = \frac{\sqrt[4]{3}}{\sqrt[4]{625}} = \frac{\sqrt[4]{3}}{5} \).

Step 4: Final Simplified Expression

The radical expression \( \sqrt[4]{\frac{3}{625}} \) simplifies to \( \frac{\sqrt[4]{3}}{5} \).

Key Concepts

Fourth Roots

When we encounter a radical expression like \( \sqrt[4]{x} \), it represents a fourth root. But what does it mean? Let's break it down. Taking the fourth root of a number is essentially asking: **What number, when multiplied by itself four times, equals the original number?**
For instance, consider the fourth root of 16, i.e., \( \sqrt[4]{16} \). If you find a number that when multiplied by itself four times gives 16, that number is 2, since \( 2 \times 2 \times 2 \times 2 = 16 \). Thus, \( \sqrt[4]{16} = 2 \).
Let's relate this back to the radical expression from the exercise: \( \sqrt[4]{625} \) simplifies to 5 because \( 5^4 = 625 \). Recognizing perfect powers in expressions helps streamline radical simplifications.

Perfect Powers

Perfect powers play a crucial role in simplifying radical expressions. A perfect power is simply a number that can be expressed as an integer raised to an exponent.
In our exercise, 625 is a perfect fourth power because it's equal to \( 5^4 \). Knowing that a number can be rewritten as an exponent helps in dealing with roots. It allows us to break down complex format numbers into simpler ones.
Some common perfect powers include:

• 16, which is \( 2^4 \)
• 81, which is \( 3^4 \)
• 256, which is \( 4^4 \)

Recognizing when denominators are perfect powers is key in simplifying expressions like \( \frac{3}{625} \) by isolating and reducing the complicating factors.

Fraction Simplification

Fraction simplification is the process of making a fraction as simple as possible by reducing the top and bottom numbers to their smallest forms. So, in terms of radical expressions, this involves splitting the radical root.
In the given problem, the fraction is inside a fourth root: \( \sqrt[4]{\frac{3}{625}} \). A helpful simplification trick here is to take the fourth root of each part separately. This allows us to write the expression as \( \frac{\sqrt[4]{3}}{\sqrt[4]{625}} \), which then simplifies to \( \frac{\sqrt[4]{3}}{5} \) because the denominator \( \sqrt[4]{625} = 5 \).
Some tips to keep in mind:

• Always look for perfect powers to simplify both the numerator and the denominator.
• Whenever possible, separate the root to individual components to simplify step by step.
• Check if dividing the root term into simpler parts makes it easier to see any further simplifications.

This method results in cleaner and more manageable expressions, crucial in further algebraic manipulations.