Problem 5 - Algebra for College Students

Question

Evaluate each numerical expression. \((-8)^{\frac{1}{3}}\)

Step-by-Step Solution

Answer

The cube root of -8 is -2.

Step 1: Understand the Expression

We need to evaluate the cube root of -8, which is expressed as \((-8)^{\frac{1}{3}}\). This means finding a number that, when multiplied by itself thrice, gives -8.

Step 2: Identify the Cube Root

The cube root of -8 is the number that satisfies \(x^3 = -8\). We start by recognizing that \((-2)\times(-2)\times(-2)=-8\), thus \((-2)^3 = -8\).

Step 3: Check Your Result

Verify that \(-2\), when cubed, indeed gives \(-8\). Calculating:\((-2)^3 = (-2) \times (-2) \times (-2) = 4 \times (-2) = -8\). This confirms our solution is correct.

Key Concepts

Understanding Negative Numbers

Negative numbers are numbers less than zero and are crucial in various mathematical contexts, including cube roots. Dealing with negative numbers requires understanding how they multiply and divide. When you multiply two negative numbers, the negatives cancel out, resulting in a positive number. However, multiplying three negative numbers, as in cube rooting, results in a negative outcome.
This is because:

• Two negatives make a positive (e.g., (-2) x (-2) = 4).
• Adding another negative multiplies back to negative (e.g., 4 x (-2) = -8).

So, when finding cube roots involving a negative, the result will be negative, as shown with (-8)^(1/3) = -2. Understanding this concept is vital for solving cube roots of negative numbers.

Exploring Exponentiation

Exponentiation is a mathematical operation involving numbers called bases and exponents. The expression egin{equation} a^{b} ewline ext{represents turning } a ext{ into } } b ext{ times multiplied version of itself}. ewline ext{When} } b ext{ equals} 1/3, it refers to the cube root of } a. ewline In the context of our problem, ewline egin{equation}(-8)^{1/3} ewline means finding a value that, when raised to the power of three, equals -8. This process calls on the properties of exponents, especially the fractional ones. Cube roots essentially "break down" } a number back into its base form, consistent with the rules of exponents:

• When a^3 = b, then a = b^{1/3}.
• This principle guides the conversion of a number to its cube root.

Understanding exponentiation's role in finding cube roots helps simplify complex expressions.

Effective Problem Solving

Problem-solving in math involves a strategic approach to understanding and tackling questions. With cube roots of negative numbers, the challenge is recognizing the relationships and rules guiding them. Effective problem-solving requires:

• Understanding the problem: Identify what you're solving. Here, it involves finding egin{equation} anumber ext{ x where } x^3 = -8. ewline
• Devising a plan: Use knowledge of negatives and exponents to find potential solutions.
• Implement the plan: Calculate or logically deduce the cube root. In our problem, recognizing the pattern that (-2)^3 = -8 helps.
• Reviewing the result: Confirm by recalculating (e.g., (-2) x (-2) x (-2) = 4 (-2) = -8 to verify correctness).

This methodical approach not only aids in solving cube roots but also enhances overall mathematical skills. It helps build confidence in tackling similar problems in the future.