Question

which monomial is a perfect cube?

• \\(1x^{10}\\)
• \\(8x^8\\)
• \\(27x^{16}\\)
• \\(9x^9\\)

Explanation:

Define perfect cube monomials

Using the Perfect Cubes and Cube Roots of Monomials knowledge points
A monomial \(ax^n\) is a perfect cube if its coefficient \(a\) is a perfect integer cube and its exponent \(n\) is a multiple of 3.

Analyze the first option

Using the Perfect Cubes knowledge point
For \(1x^{10}\):

• Coefficient: \(1 = 1^3\) (perfect cube)
• Exponent: 10 is not divisible by 3

Analyze the second option

Using the Perfect Cubes knowledge point
For \(8x^8\):

• Coefficient: \(8 = 2^3\) (perfect cube)
• Exponent: 8 is not divisible by 3

Analyze the third option

Using the Perfect Cubes knowledge point
For \(27x^{15}\):

• Coefficient: \(27 = 3^3\) (perfect cube)
• Exponent: 15 is divisible by 3 since \(15 = 3 \times 5\)
• Thus, \(27x^{15} = (3x^5)^3\)

Analyze the fourth option

Using the Perfect Cubes knowledge point
For \(9x^9\):

• Coefficient: 9 is not a perfect cube (\(9 = 3^2\))
• Exponent: 9 is divisible by 3

Answer:

• \(1x^{10}\)
• \(8x^8\)
• \(27x^{15}\) (Correct answer)
• \(9x^9\)