Problem 29 - College Algebra with Modeling and Visualization

Question

Simplify the expression. Assume that all variables are positive. $$ \sqrt[3]{8 x^{6} y^{3} z^{9}} $$

Step-by-Step Solution

Answer

The simplified expression is \( 2x^2yz^3 \).

Step 1: Understanding the Cubic Root

First, realize that the expression is a cubic root: \( \sqrt[3]{8 x^{6} y^{3} z^{9}} \). This means we are looking for a value, which when raised to the power of 3, gives the original expression.

Step 2: Extracting the Cubic Root of a Constant

Let's start with the constant, 8. The cubic root of 8 is 2 because \( 2^3 = 8 \). This means that \( \sqrt[3]{8} = 2 \).

Step 3: Extracting the Cubic Root of Each Variable

For each variable term inside the radical, take its power and divide it by 3 (the index of the root). Thus, \( \sqrt[3]{x^6} = x^{6/3} = x^2 \), \( \sqrt[3]{y^3} = y^{3/3} = y \), and \( \sqrt[3]{z^9} = z^{9/3} = z^3 \).

Step 4: Combining the Results

Combine the terms obtained from taking the cubic roots: \( 2 \times x^2 \times y \times z^3 \). So, the simplified expression is \( 2x^2yz^3 \).

Key Concepts

Cubic Root

A cubic root is an operation that finds a number which, when multiplied by itself three times, equals the original number. For the expression \( \sqrt[3]{8 x^{6} y^{3} z^{9}} \), our goal is to identify numbers or terms that result in the original value when cubed. This process is about reversing the 'cubing' of a number, making it smaller and easier to work with.

• The cubic root of a constant like 8 is 2, because \( 2^3 = 8 \).
• Understanding these roots helps simplify expressions by essentially "undoing" the cube.

Remember that taking the cubic root of a larger expression involves breaking down each component — constants and variables — to find their respective roots.

Simplification

Simplification is the process of reducing a mathematical expression into its simplest form. When simplifying algebraic expressions like \( \sqrt[3]{8 x^{6} y^{3} z^{9}} \), we focus on making the expression as straightforward as possible. This involves:

• Finding the cubic roots of constants and variables.
• Reducing the power of variables by dividing them accordingly.
• Combining the simplified components.

The simplified form of the expression, \( 2x^2yz^3 \), demonstrates simplification by converting a more complex form into one that is easily understood. Simplifying helps in solving equations faster and understanding the core of mathematical problems.

Variables

Variables are symbols that represent unknown values and can change. Commonly denoted by letters like \( x \), \( y \), and \( z \), we use them in expressions to generalize mathematical relationships. While simplifying, it's important to manipulate variables correctly to retain the expression's integrity.

• Each variable's exponent determines how the variable behaves under operations like cubing or extracting roots.
• When variables have positive values, simplifying tasks become straightforward, as assumed in the problem.

Keeping variables positive makes calculations more direct, reducing complexity in algebraic manipulations.

Powers and Exponents

Powers and exponents are key concepts in algebra that involve raising numbers or variables to a certain degree. They tell us how many times a number is multiplied by itself. For instance, in \( x^6 \), the number 6 is the exponent. When dealing with cubic roots:

• You divide the exponent by the index of the root, which is 3 in cubic roots.
• For example, \( x^6 \) becomes \( x^{6/3} = x^2 \).

This division simplifies the expression and reveals the true base as it substitutes the effect of cubing. Understanding powers and exponents is fundamental to mastering algebraic problem-solving and recognizing patterns in expressions.