Which monomial is a perfect cube? 1×3 3×3 6×3 9×3
The correct answer and explanation is:
The monomial that is a perfect cube is:
1x³
✅ Correct Answer: 1x³
🔍 Explanation (300 words):
A perfect cube is a number or expression that can be written as the cube of another number or expression. In algebraic terms, a monomial is a perfect cube if both the numerical coefficient and the variable part are perfect cubes.
Let’s analyze each choice:
1x³
- 1 is a perfect cube: 1=131 = 1^3
- x³ is a perfect cube: x3=(x)3x^3 = (x)^3
- Therefore, 1×3=(1x)31x^3 = (1x)^3
- ✅ Perfect cube
3x³
- 3 is not a perfect cube. The cube root of 3 is approximately 1.442, which is not an integer.
- Even though x³ is a perfect cube, the coefficient 3 is not.
- ❌ Not a perfect cube
6x³
- 6 is not a perfect cube: The cube root of 6 is approximately 1.817, not an integer.
- x³ is a perfect cube, but again the coefficient is not.
- ❌ Not a perfect cube
9x³
- 9 is not a perfect cube: The cube root of 9 is approximately 2.080, not an integer.
- x³ is a perfect cube, but 9 is not.
- ❌ Not a perfect cube
🔑 Key Concept:
A monomial like axnax^n is a perfect cube if and only if:
- aa is a perfect cube (like 1, 8, 27, etc.)
- nn is divisible by 3 (so that xnx^n can be written as (xk)3(x^k)^3)
Only 1x³ satisfies both conditions.