Problem 104 - Elementary and Intermediate Algebra

Question

Simplify each radical expression, if possible. Assume all variables are unrestricted. $$ \sqrt[3]{1,000 a^{6} b^{6}} $$

Step-by-Step Solution

Answer

The simplified expression is \( 10a^2b^2 \).

Step 1: Identify the Perfect Cubes

The expression is \( \sqrt[3]{1,000 a^{6} b^{6}} \). Separate it into parts. \( 1,000 = 10^3 \). \( a^6 \) and \( b^6 \) can be rewritten as \((a^2)^3\) and \((b^2)^3\) since \( a^6 = (a^2)^3 \) and \( b^6 = (b^2)^3 \).

Step 2: Apply the Cube Root

Apply the cube root to each part: \( \sqrt[3]{10^3} \), \( \sqrt[3]{(a^2)^3} \), and \( \sqrt[3]{(b^2)^3} \). The cube root of \( 10^3 \) is \( 10 \). The cube root of \( (a^2)^3 \) is \( a^2 \). Similarly, the cube root of \( (b^2)^3 \) is \( b^2 \).

Step 3: Combine the Results

Combine the simplified parts: \( 10 imes a^2 imes b^2 \). Therefore, \( \sqrt[3]{1,000 a^{6} b^{6}} = 10a^2b^2 \).

Key Concepts

Understanding Cube Roots

A cube root is a mathematical operation used to find a number that, when multiplied by itself three times, gives the original number. For example, the cube root of 8 is 2 because when you multiply 2 × 2 × 2, you get 8. In the given problem, we're looking at cube roots of expressions like \( \sqrt[3]{1,000 a^{6} b^{6}} \).
To simplify this expression, start by identifying numbers and variables that can be broken down into a cube form. Here, 1000 is a perfect cube because \( 10^3 = 1,000 \). Recognizing such relationships is key in simplifying radical expressions.

• Apply the cube root to each component separately.
• Identifying perfect cubes speeds up solving problems.

This approach not only helps simplify expressions but also strengthens your understanding of how roots function in algebraic settings.

Identifying and Using Perfect Cubes

Perfect cubes are numbers or expressions that are the result of cubing a whole number or expression. In simpler terms, if you multiply a number by itself three times, you get a perfect cube. For example, 27 is a perfect cube because \( 3^3 = 27 \).
In our problem, you identify that 1000, \( a^6 \), and \( b^6 \) can be rewritten in terms of cubic powers:

• \( 1,000 = 10^3 \)
• \( a^6 = (a^2)^3 \)
• \( b^6 = (b^2)^3 \).

Recognizing these helps to simplify the expression as you apply the cube root. Knowing these relationships is essential for simplifying and solving radical expressions quickly and accurately.
Spotting perfect cubes in algebraic terms is particularly useful when simplifying complex expressions.

Steps in Algebraic Simplification

In algebra, simplification involves reducing an expression to its simplest form. This often includes performing operations like factoring, distributing, and applying math operations like cube roots. Simplifying radical expressions like \( \sqrt[3]{1,000 a^{6} b^{6}} \) involves breaking it down into its simplest parts by utilizing known values and properties.
Follow a systematic approach:

• Identify sections of the expression that can be expressed as powers or perfect cubes.
• Apply the cube root to each recognized section.
• Combine all simplified parts to form a single, simplified expression.

In our example, this means taking \( \sqrt[3]{10^3} \), which becomes 10, \( \sqrt[3]{(a^2)^3} \) becoming \( a^2 \), and \( \sqrt[3]{(b^2)^3} \) becoming \( b^2 \), and recombining them to result in \( 10a^2b^2 \). This step-by-step simplification enhances comprehension and ensures accuracy in solving algebraic problems.