Problem 95 - Elementary Algebra

Question

Problem: Evaluate: \(\sqrt[3]{-216}\) Incorrect Answer: \(\sqrt[3]{-216}\) is not a real number.

Step-by-Step Solution

Answer

The cube root of -216 is -6.

Step 1: Understanding Cube Roots

A cube root of a number is a value that, when multiplied by itself three times, gives the original number. For example, the cube root of 8 is 2 because 2 × 2 × 2 = 8.

Step 2: Determine the Cube Root of -216

Identify the number which, when raised to the power of 3, equals -216. We need to find a number such that x^3 = -216. Since -6 × -6 × -6 = -216, the cube root of -216 is -6.

Step 3: Verify the Result

To verify, compute (-6)^3: (-6) × (-6) × (-6) = 36 × (-6) = -216. Therefore, the cube root of -216 is indeed -6.

Key Concepts

Understanding Negative Numbers

Negative numbers are numbers less than zero. They are represented with a minus sign (-) in front. For example, -1, -2, -3, etc., are negative numbers. When dealing with cube roots, it's important to understand how negative numbers behave.

Real Numbers in Cube Roots

Real numbers include all the numbers on the number line. This means they can be positive, negative, or zero. In the context of cube roots, real numbers are key because the cube root of any real number is also a real number. For example, the cube root of -216 is -6 because -6 × -6 × -6 = -216.

Verifying Cube Root Results

Verifying the results ensures you have the correct answer. In our exercise, the cube root of -216 is computed by finding a number that, when cubed, equals -216. We found that \(-6\) is the cube root because \(-6 \times -6 \times -6 = -216\). Always double-check by redoing the multiplication.