A number has 27 decimal places after the decimal point. How many decimal places will its cube root have?

Introduction / Context:
This question focuses on the relationship between decimal places of a number and its cube root. It is a concept from number system and powers of ten, frequently used when dealing with very small or very large decimal numbers.

Given Data / Assumptions:

• A decimal number has exactly 27 decimal places after the decimal point.
• We consider the cube root of this number.
• We assume the number is positive and written in standard decimal form.

Concept / Approach:
If a number is expressed as 10^(-n), then its cube root is 10^(-n/3). The number of decimal places corresponds to the magnitude of the negative exponent on 10. If a number has 27 decimal places, it behaves like 10^(-27) multiplied by some nonzero digit pattern. Taking the cube root divides the exponent by 3, so the cube root will behave like 10^(-9) times some digits, giving 9 decimal places.

Step-by-Step Solution:
Step 1: Suppose a number has 27 decimal places; for example, it can be thought of in the form k * 10^(-27), where k is not divisible by 10.Step 2: The cube root of this number is (k * 10^(-27))^(1/3).Step 3: This equals k^(1/3) * (10^(-27))^(1/3).Step 4: For powers of ten, (10^(-27))^(1/3) = 10^(-27/3) = 10^(-9).Step 5: So the cube root is some nonzero factor multiplied by 10^(-9).Step 6: A factor of 10^(-9) corresponds to 9 digits after the decimal point in standard decimal representation.Step 7: Therefore, the cube root of a number with 27 decimal places will have exactly 9 decimal places.

Verification / Alternative check:
As a simple illustration, consider the number 10^(-27) itself, which is 0.000000000000000000000000001 with 27 zeros after the decimal point before the 1. Its cube root is 10^(-9), which is 0.000000001, with 9 decimal places. This matches our reasoning.

Why Other Options Are Wrong:

• 3 and 6: These come from dividing 27 by 9 or by a wrong factor, not by 3.
• 27: This would mean the cube root has the same number of decimal places as the original number, which is not consistent with exponent rules.

Common Pitfalls:
Students sometimes confuse square roots and cube roots and may divide 27 by 2 instead of by 3. Others may think that the number of decimal places stays unchanged under roots. Understanding exponents of 10 and how roots affect them is crucial.

Final Answer:
The cube root will have 9 decimal places.