Problem 74 - College Algebra

Question

In Exercises \(69-76,\) add or subtract terms whenever possible. $$\sqrt[3]{24 x y^{3}}-y \sqrt[3]{81 x}$$

Step-by-Step Solution

Answer

The simplified form of the given expression is \(-y\sqrt[3]{3x}\).

Step 1: Simplify each term

First, break down each term inside the cube root into its prime factors: \(\sqrt[3]{24 x y^{3}}\) becomes \(\sqrt[3]{2^3*3 x y^{3}}\) and \(-y \sqrt[3]{81 x}\) becomes \(-y \sqrt[3]{3^4 x}\)

Step 2: Apply properties of radicals to simplify further

We can express \(2^3\) and \(y^{3}\) as cubes under the cube root in the first term: \(\sqrt[3]{2^3*3 x y^{3}}\) becomes \(2y\sqrt[3]{3x}\). For the second term, we can express \(3^3\) as a cube under the cube root and move the remaining factor of 3 outside of the cube root: \(-y\sqrt[3]{3^4x}\) becomes \(-3y\sqrt[3]{3x}\).

Step 3: Combine like terms

Now the two terms can be combined because they have the same cube root part: \(2y\sqrt[3]{3x} - 3y\sqrt[3]{3x}\) simplifies to \(-y\sqrt[3]{3x}\).

Key Concepts

Radicals in Algebra

In algebra, a radical is used to represent the root of a number, which tells you how to 'undo' an exponent. The cube root, which is denoted by the symbol \(\sqrt[3]{}\), is a type of radical that asks the question: Which number, when multiplied by itself three times, gives us the original number? For example, \(\sqrt[3]{8}\) equals 2, because \(2 \times 2 \times 2 = 8\).

In the exercise \(\sqrt[3]{24xy^3} - y\sqrt[3]{81x}\), the first step towards simplifying the expression is to express the radicands, the numbers under the radical sign, using their prime factors. Understanding how to manipulate these radicals is fundamental in algebra, particularly when simplifying expressions or solving equations. It's all about transforming the provided expressions into a simpler or more useful form using the properties of radicals, which will be discussed in the next sections.

Combining Like Terms

An important concept in algebra is 'combining like terms'. This refers to the process of simplifying algebraic expressions by adding or subtracting terms that have the same variable factors and are raised to the same power. In our example with cube roots, like terms are terms that have the same cubed variable after simplification.

For instance, in the expression \(2y\sqrt[3]{3x} - 3y\sqrt[3]{3x}\), after simplifying the cube roots, we notice that both terms include the \(\sqrt[3]{3x}\) and are therefore like terms. To combine them, we simply perform the operations on the coefficients in front of the radical, resulting in \( (-3 + 2)y\sqrt[3]{3x}\) or \( -y\sqrt[3]{3x}\). The ability to recognize and combine like terms is crucial for simplification and to make more complex calculations manageable.

Properties of Radicals

Understanding the properties of radicals is key to simplifying expressions that contain roots, including square roots, cube roots, and higher order roots. These properties include the product and quotient rules for radicals, which allow us to multiply and divide within the radical, as well as the ability to take out powers that match the type of root.

In the step-by-step solution provided, the expression \(\sqrt[3]{2^3*3xy^3}\) takes advantage of the product property. Since \(2^3\) and \(y^3\) are perfect cubes, they can be taken out of the cube root as \(2\) and \(y\). Furthermore, for the expression \(\sqrt[3]{3^4x}\) the quotient property is applied, leaving a single 3 outside of the radical. These properties permit the transformation of radicals into simpler forms that maintain the original value of the expression, ultimately leading to a final solution that is easier to grasp and use for further mathematical operations.