In this number classification question, the numbers 64, 162, 27 and 144 are given. Three of these numbers are perfect powers (either perfect squares or perfect cubes of integers). Which number is different from the others in this group?

Introduction / Context:
This classification question involves recognising special properties of numbers, particularly perfect squares and perfect cubes. You are given the numbers 64, 162, 27 and 144. Three of them can be expressed as an exact integer power (square or cube) of a positive integer, while one cannot. The task is to determine which number does not fit the pattern of being a perfect square or cube.

Given Data / Assumptions:
• The numbers to compare are 64, 162, 27 and 144.• A perfect square is a number of the form n^2 for some integer n.• A perfect cube is a number of the form n^3 for some integer n.• We check which numbers can be written exactly as such powers.

Concept / Approach:
We test each number to see if it is a perfect square or a perfect cube. 64 is a well known perfect power: 64 = 4^3 and also 64 = 8^2. The number 27 is 3^3, a perfect cube. The number 144 is 12^2, a perfect square. Now we examine 162. It can be factorised as 2 * 81 = 2 * 3^4, but there is no integer n such that n^2 = 162 or n^3 = 162. That means 162 is not a perfect square or perfect cube, while the other three numbers are. Therefore, 162 is the odd one out.

Step-by-Step Solution:
Step 1: Test 64. We know 8 * 8 = 64, so 64 is 8^2, a perfect square. Also, 4 * 4 * 4 = 64, so 64 is also 4^3, a perfect cube. It is clearly a perfect power.Step 2: Test 27. We know 3 * 3 * 3 = 27, so 27 is 3^3, a perfect cube.Step 3: Test 144. We know 12 * 12 = 144, so 144 is 12^2, a perfect square.Step 4: Test 162. Factor 162: 162 = 2 * 81 = 2 * 3^4. The square root of 162 is not an integer (it lies between 12 and 13), and the cube root is also not an integer (it lies between 5 and 6).Step 5: Therefore 162 is not a perfect square or perfect cube.Step 6: Since three numbers (64, 27 and 144) are perfect powers, and 162 is not, 162 is the odd one out.

Verification / Alternative check:
We can further verify by checking nearby squares and cubes. For squares, 12^2 = 144 and 13^2 = 169, so 162 is in between and cannot be a perfect square. For cubes, 5^3 = 125 and 6^3 = 216, so 162 lies between them and cannot be a perfect cube. No other simple integer power equals 162. This confirms that 162 does not share the perfect square or cube property that the other three numbers have.

Why Other Options Are Wrong:
64: A perfect square (8^2) and also a perfect cube (4^3), so it definitely fits the pattern.27: A perfect cube (3^3), also fitting the pattern.144: A perfect square (12^2), again fitting the pattern.None of these: Incorrect, because we clearly identified 162 as not being a perfect square or cube.

Common Pitfalls:
Some students may be distracted by the fact that 162 is not as commonly memorised as a standard square or cube and might suspect another number instead. The safe method is to check each number carefully with nearby squares and cubes. Once you verify that 64, 27 and 144 are exact powers and 162 is not, the classification becomes clear: 162 is the odd one out.

Final Answer:
The number that is different from the others is 162.