Problem 65 - Intermediate Algebra

Question

Use the quotient rule to divide. Then simplify if possible. See Example 5 $$ \frac{5 \sqrt[4]{48}}{\sqrt[4]{3}} $$

Step-by-Step Solution

Answer

The simplified expression is 10.

Step 1: Understanding the Problem

We need to divide two radical expressions: \( \frac{5 \sqrt[4]{48}}{\sqrt[4]{3}} \). First, let's understand that we are dealing with fourth roots in both the numerator and denominator.

Step 2: Apply the Quotient Rule for Radicals

The quotient rule for radicals states that \( \frac{\sqrt[n]{a}}{\sqrt[n]{b}} = \sqrt[n]{\frac{a}{b}} \). Apply this rule to the expression to simplify: \( 5 \sqrt[4]{\frac{48}{3}} \).

Step 3: Simplify the Radical Expression Inside the Root

Calculate \( \frac{48}{3} \) which results in \( 16 \). Thus, the expression becomes \( 5 \sqrt[4]{16} \).

Step 4: Evaluate the Fourth Root

Evaluate the fourth root of 16. Since \( 16 = 2^4 \), the fourth root of 16 is 2. Thus, our expression now simplifies to \( 5 \times 2 \).

Step 5: Simplify the Expression

Multiply 5 by 2 to obtain the final simplified result: 10.

Key Concepts

Simplification of Radicals

The process of simplifying radicals is all about reducing the expression to its simplest form. When simplifying radicals like \( \sqrt[n]{a} \), we want to find if \( a \) can be broken down into its prime factors that can be extracted from the root.

Here's how to simplify radicals step by step:

• First, check if the radicand (the number inside the radical) is a perfect power of the root. For instance, the fourth root of 16 is straightforward since \( 16 = 2^4 \), making it exactly \( 2 \).
• If the number isn't a perfect power, break it down into prime factors to see if any can combine under the root. For example, \( \sqrt[4]{48} \) can be broken down into \( 2^4 \times 3 \). Here, you can take \( 2 \) out of the radical and leave 3 inside, resulting in \( 2\sqrt[4]{3} \).

Simplification not only makes expressions cleaner but often helps reveal underlying relationships or patterns in the numbers.

Evaluating Radical Expressions

Evaluating radical expressions involves finding the actual number that the radical represents. This task becomes easier once the radicals are simplified.

For example, consider the expression \( 5 \sqrt[4]{16} \):

• After simplifying the radical \( \sqrt[4]{16} \) (since \( 16 = 2^4 \)), we know that it evaluates to \( 2 \).
• So, to evaluate the entire expression, we multiply by the coefficient, which is 5 in our case. Hence, the expression becomes \( 5 \times 2 = 10 \).

Evaluating radicals helps us understand the numeric value of the expression, thus allowing us to solve problems more effectively. Remember, simplification is usually the vital first step before evaluation.

Fourth Roots

Fourth roots are a kind of radical where the index is 4, meaning we are looking for a number that, when multiplied by itself four times, gives the original number. The notation \( \sqrt[4]{a} \) tells us we are considering the fourth root of \( a \).

Understanding how to find the fourth root is crucial:

• A number like 16 is straightforward since, as reasoned before, \( 16 = 2^4 \), so \( \sqrt[4]{16} = 2 \).
• For more complex numbers, you might first need to breakdown the number into its prime factors to detect any perfect fourth powers.

Mastering fourth roots helps in simplifying expressions early in the process, making it vital for smoothing out equations and solutions that involve higher order roots.