Problem 82 - Elementary and Intermediate Algebra

Question

Simplify each cube root. See Example \(6 .\) $$ \sqrt[3]{-343 a^{6} b^{3}} $$

Step-by-Step Solution

Answer

The simplified expression is \(-7a^2b\).

Step 1: Recognize the Cube Root Expression

The given expression is \( \sqrt[3]{-343 a^{6} b^{3}} \). Our goal is to simplify this cube root expression by breaking it down into its factors.

Step 2: Simplify the Cube Root of Numerical Part

Identify the cube root of the numerical part. We need to find \( \sqrt[3]{-343} \). Since \( (-7)^3 = -343 \), it follows that \( \sqrt[3]{-343} = -7 \).

Step 3: Simplify the Cube Root of Variable Part

For the variable part \( a^6 b^3 \), we apply the cube root: 1. For \( a^6 \), we get \( (a^2)^3 = a^6 \), so \( \sqrt[3]{a^6} = a^2 \).2. For \( b^3 \), \( (b)^3 = b^3 \), so \( \sqrt[3]{b^3} = b \).

Step 4: Combine Simplified Parts

Combine the simplified results: - From Step 2: \( -7 \)- From Step 3: \( a^2 \) and \( b \) Putting these together, the simplification of the cube root is \( -7a^2b \).

Key Concepts

Numerical Simplification

When faced with a cube root expression involving numbers, like \( \sqrt[3]{-343} \), the first step is to simplify the numeric part independently. Numerical simplification is all about breaking down the number inside the cube root into a simpler or more understandable form. We know that the cube root of a number \( x \) is a value \( y \) such that \( y^3 = x \). In our example, \(-343\) is the number we’re simplifying.

• - You can simplify \( \sqrt[3]{-343} \) by looking for a number that, when cubed, equals \(-343\).
• Since \((-7)^3 = -343\), it follows that \( \sqrt[3]{-343} = -7 \).

Breaking down these calculations shows us how this process works, helping us find that \( -7 \) is the cube root of \( -343 \). This simplification step focuses on the mathematical power of decomposing a cubic equation back to its base.

Variable Factorization

Now, let's explore variable factorization. In simplifying cube roots, we must also consider algebraic expressions involving variables, such as \( a^6 b^3 \). Factorization helps us identify and separate powers of variables so that we can apply the cube root.

• Look at \( a^6 \): Identifying a perfect cube, using \( a^6 = (a^2)^3 \), tells us the cube root is \( a^2 \).
• Similarly, for \( b^3 \): Since \( b^3 \) is already a perfect cube, the cube root is simply \( b \), because \( (b)^3 = b^3 \).

The power of variable factorization lies in recognizing these cubes and simplifying them. This process makes it easier to work with complex algebraic expressions and find their cube roots efficiently.

Algebraic Expression Simplification

Algebraic expression simplification is a broader concept that encompasses the simplification of whole expressions, involving both numeric and variable parts like \( \sqrt[3]{-343 a^6 b^3} \). The goal is to express the original complex expression in its simplest and most comprehensible form.

• From our previous steps, we found that \( \sqrt[3]{-343} = -7 \).
• We simplified the variable part to \( a^2b \).
• Combining both, \( -7a^2b \) becomes the simplified form.

This final step of combining the numeric and variable simplifications allows us to arrive at a simple, easy-to-understand expression that accurately represents the original cube root. Simplification reduces complexity and aids our understanding, allowing us to handle algebraic expressions efficiently in any mathematical context.