Problem 60 - College Algebra with Corequisite Support

Question

For the following exercises, simplify each expression. \(\sqrt[3]{24 x^{6}}+\sqrt[3]{81 x^{6}}\)

Step-by-Step Solution

Answer

The simplified expression is \(5\cdot \sqrt[3]{3}x^2\).

Step 1: Break Down the Expression

The expression to simplify is \(\sqrt[3]{24 x^{6}}+\sqrt[3]{81 x^{6}}\). We will address each cube root separately.

Step 2: Simplify Cube Roots

For \(\sqrt[3]{24 x^{6}}\), notice that it can be split into \(\sqrt[3]{8} \cdot \sqrt[3]{3} \cdot \sqrt[3]{x^6}\). Similarly, for \(\sqrt[3]{81 x^{6}}\), it can be split into \(\sqrt[3]{27} \cdot \sqrt[3]{3} \cdot \sqrt[3]{x^6}\).

Step 3: Simplify Expressions Involving Perfect Cubes

Since \(\sqrt[3]{8} = 2\) and \(\sqrt[3]{27} = 3\), we rewrite the expression: \(2\cdot \sqrt[3]{3}\cdot x^2 + 3\cdot \sqrt[3]{3}\cdot x^2\).

Step 4: Combine Like Terms

Since both parts are multiples of \(\sqrt[3]{3}x^2\), combine them: \((2 + 3)\cdot \sqrt[3]{3}\cdot x^2 = 5\cdot \sqrt[3]{3}\cdot x^2\).

Key Concepts

Understanding Cube Roots

The cube root, symbolized by \(\sqrt[3]{}\), represents a number that, when multiplied by itself three times, gives the original number. For instance, the cube root of 8 is 2, because \(2 \times 2 \times 2 = 8\). Similarly, the cube root of 27 is 3, as \(3 \times 3 \times 3 = 27\).

When simplifying expressions that involve cube roots, it often helps to break down the number inside the root into its factors. For example, \(\sqrt[3]{24} = \sqrt[3]{8} \cdot \sqrt[3]{3}\) because 24 can be expressed as \(8 \times 3\). By identifying factors that are perfect cubes, like 8 and 27, we can simplify the expression more easily.

Perfect Cubes and Their Use

Perfect cubes are numbers that can be written as the product of an integer multiplied by itself three times. Numbers like 1, 8, 27, and 64 are perfect cubes. Recognizing these numbers is vital when simplifying expressions involving cube roots.

For example, knowing \(\sqrt[3]{8} = 2\) helps streamline the simplification process. Recognizing perfect cubes is a skill that comes with practice, and it can help tackle complex algebraic expressions more effectively.

The Basics of Algebraic Expressions

Algebraic expressions consist of variables, constants, and operations such as addition, subtraction, multiplication, and division. In our exercise, \(\sqrt[3]{24 x^{6}}\) and \(\sqrt[3]{81 x^{6}}\) are algebraic expressions because they include variables \(x\) raised to a power.

When dealing with such expressions, simplify each part separately. Factorizing numbers and handling exponents carefully can help in combining terms efficiently. For instance, \(x^6\) can be broken into \((x^2)^3\), thus making the simplification process easier with cube roots.

Combining Like Terms

Combining like terms is a crucial skill in algebra. It involves adding or subtracting terms that have similar characteristics such as the same variable raised to the same power. In this exercise, both parts of the expression involved \(\sqrt[3]{3}x^2\).

By recognizing \(2 \cdot \sqrt[3]{3}x^2\) and \(3 \cdot \sqrt[3]{3}x^2\) as like terms, we add their coefficients — 2 and 3 — resulting in \(5 \cdot \sqrt[3]{3}x^2\).

• Step 1: Identify coefficients.
• Step 2: Add or subtract coefficients of like terms.
• Step 3: Reattach the common variable and/or operation.

This process simplifies complex expressions and makes calculations straightforward.