Problem 26 - College Algebra with Modeling and Visualization

Question

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

Step-by-Step Solution

Answer

The simplified expression is \( 2y \sqrt[3]{x} \).

Step 1: Identify the Cube Root

The expression we need to simplify is \( \sqrt[3]{8xy^3} \). A cube root simplifies the components inside by pulling out anything that can be expressed as a cube, i.e., anything of the form \( a^3 \).

Step 2: Find Cube Roots of Numerical and Variable Parts

First, consider the number 8 inside the cube root. Notice that \( 8 = 2^3 \), so its cube root is 2. Next, consider the variables. The variable \( y^3 \) is already a perfect cube, so its cube root is \( y \). The variable \( x \) is not a perfect cube and remains under the cube root.

Step 3: Combine the Simplified Parts

Combine the results from the previous step. The cube root of 8 is 2, and the cube root of \( y^3 \) is \( y \). Therefore, \( \sqrt[3]{8 x y^3} = 2y \cdot \sqrt[3]{x} \).

Key Concepts

Cube Roots

A cube root is a special type of root used to simplify expressions, particularly when dealing with perfect cubes. It is the number that, when multiplied by itself three times, yields the original number. In mathematical terms, if you have a number or expression under a cube root, like \( \sqrt[3]{a} \), it is equivalent to finding the number \( b \) such that \( b^3 = a \).
When simplifying cube roots:

• Identify if the number or variable can be expressed as a power of three, such as \( x^3 \).
• Pull out the cube root of numbers or expressions that are perfect cubes.
• What remains under the root will generally be parts of the expression that cannot be simplified into a complete cube.

In our given expression, \( 8 \) is easily recognizable as a perfect cube \( (2^3) \), making its cube root straightforward to find.

Variables in Algebra

In algebra, variables like \( x \) and \( y \) are symbols used to represent unknown or changeable values. An expression like \( xy^3 \) involves variables multiplied together, sometimes with numerical coefficients. When simplifying expressions with cube roots, it's important to identify if the power of any variable is a perfect cube, which would allow you to take its cube root directly.
Here are some steps to consider:

• Look for powers of variables that are multiples of three, such as \( y^3 \), because they indicate perfect cubes.
• Extract them from under the cube root, simplifying your expression.
• Leave variables that are not perfect cubes under the cube root, like \( x \) in our original expression.

Understanding how to manage variables effectively is key to simplifying algebraic expressions with cube roots.

Perfect Cubes

Perfect cubes are numbers or expressions in mathematics that can be expressed as \( a^3 \), where \( a \) is an integer. Recognizing these in an expression helps tremendously in simplifying under cube roots.
For example, numbers such as 8, 27, and 64 are all perfect cubes because:

• \( 8 = 2^3 \)
• \( 27 = 3^3 \)
• \( 64 = 4^3 \)

When looking at variables, powers that are multiples of three, like \( y^3 \), \( x^6 \), etc., are perfect cubes, making them simple to extract when simplifying cube roots.
In the expression \( \sqrt[3]{8xy^3} \), both 8 and \( y^3 \) are perfect cubes, thus their cube roots (\( 2 \) and \( y \) respectively) are easily extracted, resulting in a simplified form.