Problem 60 - College Algebra

Question

Simplify each expression. $$\sqrt[3]{24 x^{6}}+\sqrt[3]{81 x^{6}}$$

Step-by-Step Solution

Answer

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

Step 1: Identify the Cube Roots

First, identify the cube roots given in the expression: \(\sqrt[3]{24x^6}\) and \(\sqrt[3]{81x^6}\). Our goal is to simplify each term separately.

Step 2: Simplify \(\sqrt[3]{24x^6}\)

To simplify \(\sqrt[3]{24x^6}\), start by breaking down 24 into its prime factors: \(24 = 2^3 \times 3\). This can be rewritten as \(\sqrt[3]{2^3 \times 3 \times (x^6)}\). Since the cube root of \(2^3\) is 2, and \(x^6\) becomes \((x^2)^3\) which simplifies to \(x^2\), the cube root of this expression simplifies to \(2x^2\sqrt[3]{3}\).

Step 3: Simplify \(\sqrt[3]{81x^6}\)

Next, simplify \(\sqrt[3]{81x^6}\). Break down 81 into prime factors: \(81 = 3^4\). Using \(\sqrt[3]{3^3 \times 3 \times (x^6)}\), the cube root of \(3^3\) is 3, and \(x^6\) becomes \((x^2)^3\) simplifying to \(x^2\). Therefore, the cube root is \(3x^2\sqrt[3]{3}\).

Step 4: Combine the Simplified Terms

Now that we have simplified each cube root, combine them together: \(2x^2\sqrt[3]{3} + 3x^2\sqrt[3]{3}\). Because both terms have the common factor \(x^2\sqrt[3]{3}\), they can be combined, resulting in \((2 + 3)x^2\sqrt[3]{3}\).

Step 5: Final Expression

Adding the coefficients from the previous step, the result is \(5x^2\sqrt[3]{3}\). This is the simplified expression of the original problem.

Key Concepts

Cube Roots

Understanding cube roots can be much easier than you might think. A cube root is a number that, when multiplied by itself three times, gives you the original number. It's like asking "what number cubed equals this number?"

• For example, the cube root of 27 is 3 because 3 × 3 × 3 = 27.
• Notation: We express the cube root of a number \(a\) as \(\sqrt[3]{a}\).

Cube roots play a crucial role in simplifying radical expressions, especially when they involve variables. In the problem, we find cube roots of expressions like \(\sqrt[3]{24x^6}\), which involves both numbers and algebraic terms.

Prime Factorization

Prime factorization is an essential skill in simplifying expressions involving cube roots. It breaks down a number into its basic "building blocks" - the prime numbers.

• To factor a number like 24, you would write: 24 = 2 \(\times\) 2 \(\times\) 2 \(\times\) 3, or \(2^3 \times 3\).
• This helps to identify complete cubes within the number, which are crucial in extracting cube roots.

Once you factor the number, you look for groups of three of the same prime. For instance, with \(24 = 2^3 \times 3\), the cube root of \(2^3\) is 2. That step simplifies the expression significantly.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operators. They are foundational in algebra and crucial for solving problems like simplifying cube roots.

• In the expression \(\sqrt[3]{24x^6}\), the algebraic part is \(x^6\).
• For simplification, understanding how to break down and manipulate these expressions is key.

When dealing with cube roots, you often encounter expressions like \((x^2)^3\), which allow for simplifying to \(x^2\) outside the cube root. This is similar to factoring numbers but with variables.

Exponents

Exponents are power tools in mathematics, showing how many times a number is multiplied by itself. They appear frequently in expressions with cube roots and add a layer of complexity.

• Exponents are notated as \(b^n\), meaning the base \(b\) raised to the power of \(n\).
• In our exercise, \(x^6\) means \(x\) is multiplied by itself six times.

Understanding how exponents interact with cube roots is crucial to simplifying expressions. For example, if you have \(x^6\), this can be rewritten as \((x^2)^3\) when working with cube roots. This allows \(x^2\) to be factored out of the cube root, drastically simplifying the expression.