Problem 16 - College Algebra with Modeling and Visualization

Question

Simplify the expression. Assume that all variables are positive. $$ \sqrt[5]{16} \cdot \sqrt[5]{-2} $$

Step-by-Step Solution

Answer

The simplified expression is \(-2\).

Step 1: Identify the Expression

We need to simplify the product of two fifth roots: \( \sqrt[5]{16} \cdot \sqrt[5]{-2} \). Let us first express these roots separately.

Step 2: Apply the Properties of Radicals

According to the property \( \sqrt[n]{a} \cdot \sqrt[n]{b} = \sqrt[n]{a \cdot b} \), we can combine the two expressions as follows: \( \sqrt[5]{16} \cdot \sqrt[5]{-2} = \sqrt[5]{16 \cdot (-2)} \).

Step 3: Perform the Multiplication Inside the Radical

Calculate the product of 16 and -2: \( 16 \cdot (-2) = -32 \). Thus, the expression becomes \( \sqrt[5]{-32} \).

Step 4: Simplify the New Expression

Since \( -32 = -2^5 \), we can rewrite the expression as \( \sqrt[5]{-2^5} \). Therefore, the simplified result is \( -2 \).

Key Concepts

Properties of Radicals

Radicals, or roots, are mathematical symbols used to determine what number, when multiplied by itself a certain number of times, equals a given value. When working with radicals, it's essential to understand some rules or properties that help simplify and manipulate expressions.

One important property is that you can multiply radicals of the same degree by combining their radicands. In other words, if you have two numbers under the same root, such as \( \sqrt[n]{a} \) and \( \sqrt[n]{b} \), you can compute \( \sqrt[n]{a \cdot b} \). This property helps in simplifying expressions and makes calculations easier.

• When applying this, remember the radicals should be of the same degree.
• Only real numbers should be considered unless specified otherwise (here we assume all variables are positive).
• This property only holds for the multiplication or division of radicals.

Understanding and applying these properties efficiently is key to simplifying radical expressions.

Simplification of Expressions

Simplifying radical expressions involves various strategies to make them more manageable or to find a simpler equivalent. In the context of radicals, simplification often means expressing the radicals in their simplest form, particularly looking for common factors that can be taken out from under the root.

For example, to simplify \( \sqrt[5]{16 \cdot (-2)} \), combine them under a single radical using the property discussed earlier. Once combined, perform the arithmetic operations inside the radical before proceeding.

Here's what you do:

• Multiply the values inside to find the new radicand.
• In this case, \( 16 \times (-2) = -32 \).
• If possible, express the radicand as a power of a whole number to simplify the radical further.

You want to aim for seeing patterns or powers within the radicand that correspond to the root, as it simplifies the expression further, such as recognizing \( -32 \) as \( -2^5 \) in the given problem.

Fifth Roots

Fifth roots involve finding a number that, when multiplied by itself five times, results in the given number. Fifth roots, like other roots, have their properties and can be simplified similarly to square or cube roots.

To simplify an expression like \( \sqrt[5]{-32} \), the key is to recognize the composition of the number under the radical. Here, \( -32 \) is actually \( -2^5 \). Thus, the fifth root of \( -32 \) is simply \( -2 \), because \( (-2)^5 = -32 \).

Key aspects to consider:

• Look for the base number raised to the fifth power.
• The fifth root of a number can be positive or negative, depending on its even or odd nature.
• Recognize that negative numbers are possible for odd roots because multiplying an odd number of negative factors results in a negative number.

By grasping how to identify and calculate fifth roots, computational skills become much more intuitive.