Problem 115 - Elementary and Intermediate Algebra

Question

The following problems involve addition, subtraction, and multiplication of radical expressions, as well as rationalizing the denominator. Perform the operations and simplify, if possible. All variables represent positive real numbers. $$ \sqrt[4]{3 n^{2}} \sqrt[4]{27 n^{3}} $$

Step-by-Step Solution

Answer

The simplified expression is \( 3n\sqrt[4]{n} \).

Step 1: Apply the Product Property of Radicals

The product property of radicals states \( \sqrt[n]{a} \cdot \sqrt[n]{b} = \sqrt[n]{a \cdot b} \). Apply this to the given expression: \( \sqrt[4]{3n^2} \cdot \sqrt[4]{27n^3} = \sqrt[4]{3n^2 \cdot 27n^3} \).

Step 2: Simplify the Inside of the Radicals

Multiply the terms inside the radical: \( 3n^2 \cdot 27n^3 = 81n^5 \). So, we have \( \sqrt[4]{81n^5} \).

Step 3: Simplify Using Perfect Powers

Notice that \( 81 \) is \( 3^4 \), which is a perfect fourth power, and rewrite \( n^5 \) as \( n^4 \cdot n \). The expression becomes \( \sqrt[4]{3^4n^4n} \).

Step 4: Take Out Perfect Powers

Using the property \( \sqrt[n]{a^n} = a \), extract the perfect powers: \( \sqrt[4]{3^4} = 3 \) and \( \sqrt[4]{n^4} = n \). Thus, \( \sqrt[4]{81n^5} = 3n\sqrt[4]{n} \).

Step 5: Complete the Simplification

The simplified form of the expression \( \sqrt[4]{3n^2} \cdot \sqrt[4]{27n^3} \) is \( 3n\sqrt[4]{n} \).

Key Concepts

Product Property of Radicals

When dealing with radicals in multiplication, the Product Property of Radicals is your best friend. This property allows you to multiply two radicals together by joining them under a common radical sign. If you have radicals with the same index, like square roots or fourth roots, you can combine them easily.
For instance, let's take two fourth roots:

• \( \sqrt[4]{a} \)
• \( \sqrt[4]{b} \)

By applying the property, they can be written as a single radical:

• \( \sqrt[4]{a} \cdot \sqrt[4]{b} = \sqrt[4]{a \cdot b} \)

This simplification makes it easier to work with and understand the relationship between the terms. The key is that the radicals must have the same index for this property to be applicable.

Simplification of Radicals

Simplifying radicals involves reducing them to their simplest form, which often means removing any perfect powers from under the radical sign. This is done to make the expression cleaner and easier to work with.
Let's simplify \( \sqrt[4]{81n^5} \). First, notice the number under the radical:

• Identify perfect powers: \( 81 \) can be written as \( 3^4 \), and among \( n^5 \), \( n^4 \) is a perfect fourth power.
• Rewrite the expression: So, \( \sqrt[4]{81n^5} \) becomes \( \sqrt[4]{3^4n^4n} \).

Next, use the property \( \sqrt[n]{a^n} = a \) to simplify further:

• Extract perfect powers: Take out \( 3 \) and \( n \), giving \( 3n\sqrt[4]{n} \).

Simplification is all about rewriting the expression by taking advantage of the perfect powers, making it more manageable.

Rationalizing the Denominator

Rationalizing the denominator is a technique used to eliminate radicals from the bottom part of a fraction. While the given step-by-step does not involve a fraction, understanding this process is beneficial for broader exercises.
Whenever a radical appears in a denominator, the goal is to remove it, achieving a rational number. Here's a basic idea on how to do it:

• Multiply by a conjugate: If you have a simple square root, multiply both the numerator and the denominator by that radical. This results in a rational number in the denominator.
• Extend to higher roots: For more complex roots, multiply by the appropriate power to get rid of the radical (e.g., if you have \( \sqrt[4]{a} \), multiply by \( \sqrt[4]{a^3} \) to make \( a \) appear).

By practicing this, you ensure all terms in the expression are rational, making it easier to perform other operations.