What is the cube root of 0.000216?

Introduction / Context:
This is a basic question on cube roots of decimal numbers. It asks for the cube root of 0.000216, which can be handled either by converting the decimal to a fraction or by using known cube values and decimal place rules.

Given Data / Assumptions:
- We need ∛0.000216.
- The given options are 0.6, 0.06, 0.77 and 0.87.

Concept / Approach:
We recall that 216 is a perfect cube, since 6³ = 6 * 6 * 6 = 216. Then we use the idea that moving the decimal point in the original number corresponds to adjusting the decimal point in its cube root. Expressing 0.000216 in scientific notation helps in understanding how the cube root acts on powers of 10.

Step-by-Step Solution:
Step 1: Write 0.000216 as 216 / 10⁶, because 0.000216 = 216 divided by 1,000,000.Step 2: Recognize that 216 = 6³.Step 3: Also note that 10⁶ = (10²)³, since (10²)³ = 10⁶.Step 4: Therefore, 0.000216 = 216 / 10⁶ = 6³ / (10²)³ = (6 / 10²)³.Step 5: This shows that the cube root of 0.000216 is 6 / 10² = 6 / 100 = 0.06.

Verification / Alternative check:
Compute 0.06³ directly: 0.06 * 0.06 = 0.0036, and 0.0036 * 0.06 = 0.000216. This confirms that 0.06 is indeed the cube root of 0.000216.

Why Other Options Are Wrong:
0.6³ equals 0.216, which is 1000 times larger than 0.000216. The options 0.77 and 0.87 are not simple fractions of known cubes and their cubes would be well above 0.000216. Therefore they do not match the required cube root.

Common Pitfalls:
A common mistake is to treat decimal cube roots as if they were square roots and move the decimal point incorrectly. Another error is ignoring the exponent on 10 when splitting the number into mantissa and power of 10. Using scientific notation or writing the decimal as a fraction helps avoid these errors.

Final Answer:
The cube root of 0.000216 is 0.06.