Problem 8 - Intermediate Algebra

Question

Add or subtract. $$ 2 \sqrt[3]{3 a^{4}}-3 a \sqrt[3]{81 a} $$

Step-by-Step Solution

Answer

\(-7a \sqrt[3]{3a}\)

Step 1: Simplify the Cube Roots

First, simplify each cube root separately. For the term \( \sqrt[3]{3a^4} \), note there is nothing we can simplify further as \( 3a^4 \) is already broken down into its basic parts.For the term \( \sqrt[3]{81a} \), recognize that \( 81 = 3^4 \), thus rewrite it as \( \sqrt[3]{3^4a} \). Simplify to \( 3 \cdot \sqrt[3]{3a} \) because \( \sqrt[3]{3^3} = 3 \). So the term \( -3a \cdot \sqrt[3]{81a} \) simplifies to \( -3a \cdot 3 \cdot \sqrt[3]{3a} = -9a \sqrt[3]{3a} \).

Step 2: Factor to Combine Terms

Notice both terms now involve \( \sqrt[3]{3a} \). The expression becomes: \( 2 \sqrt[3]{3a^4} - 9a \sqrt[3]{3a} \).Extract \( \sqrt[3]{3a} \) as a common factor. The expression can be factored as:\( (2 \sqrt[3]{a^3} - 9a) \sqrt[3]{3a} \).

Step 3: Simplify the Factor Inside the Parentheses

Simplify \( \sqrt[3]{a^3} \) to \( a \), as \( (a^3)^{1/3} = a \). Now the expression inside the parentheses becomes:\( (2a - 9a) \).Therefore, \( 2a - 9a = -7a \).

Step 4: Write the Final Expression

Substitute back into the factored form:The expression is \( -7a \cdot \sqrt[3]{3a} \). Thus, the final answer is:\( -7a \sqrt[3]{3a} \).

Key Concepts

Cube Roots

Understanding cube roots can make solving problems a lot easier. A cube root is a special value that, when multiplied by itself three times, gives the original value. For example, the cube root of 27 is 3 because \( 3 \times 3 \times 3 = 27 \). We represent cube roots with the symbol \( \sqrt[3]{ } \). When you encounter expressions like \( \sqrt[3]{81a} \), break it down into smaller parts.

• Recognize perfect cubes: For example, 81 is equivalent to \( 3^4 \), and one of the factors, \( 3^3 \), comes out of the cube root as 3.
• Leave any leftover factors inside the cube root: In this example, \( a \) remains inside as \( \sqrt[3]{3a} \).

This simplification helps in combining terms or factoring later. Remember that not all numbers will simplify nicely, but breaking them down may still reveal valuable insights.

Factoring Expressions

Factoring is a powerful tool in algebra that simplifies expressions by identifying common elements within them. It can transform a complex expression into a more manageable one by focusing on shared components. To factor properly:

• Look for common factors: Check if different terms share a component that can be factored out.
• For the example \( 2 \sqrt[3]{3a^4} - 9a \sqrt[3]{3a} \), notice both terms include \( \sqrt[3]{3a} \).

By factoring out \( \sqrt[3]{3a} \), the expression simplifies to \((2 \sqrt[3]{a^3} - 9a) \sqrt[3]{3a} \). Factoring expressions not only streamlines calculations but also reveals hidden symmetries or patterns within the problem.

Combining Like Terms

Combining like terms is a key algebraic technique used to simplify expressions. This involves grouping together terms that have the same variable raised to the same power. Simplifying by combining turns a complex expression into a more straightforward form.For instance, after factoring out \( \sqrt[3]{3a} \), we ended up with \( 2a - 9a \) inside parentheses. These are like terms because they both have \( a \) raised to the first power. You simply perform the arithmetic:

• Add or subtract the coefficients of like terms.
• In this case, it becomes \( 2a - 9a = -7a \).

Once simplified, it affects the overall expression when re-incorporating the factor, transforming the work into a more elegant solution, such as \( -7a \cdot \sqrt[3]{3a} \). This process not only declutters the problem but ensures accuracy in achieving the simplest form possible.