Answer :
To determine which monomial is a perfect cube, we need to check each option to see if it can be expressed as a power of three.
A monomial is a perfect cube when both the numerical coefficient and the variable's exponent are perfect cubes.
Here's how to solve it step-by-step:
1. Understand the Problem:
We have four monomials: [tex]\(1x^3\)[/tex], [tex]\(3x^3\)[/tex], [tex]\(6x^3\)[/tex], and [tex]\(9x^3\)[/tex]. We need to check if their coefficients are perfect cubes.
2. Check Each Coefficient:
- For [tex]\(1x^3\)[/tex]:
- The coefficient is 1. The cube root of 1 is [tex]\(1\)[/tex], since [tex]\(1^3 = 1\)[/tex].
- Thus, 1 is a perfect cube.
- For [tex]\(3x^3\)[/tex]:
- The coefficient is 3. The cube root of 3 is not an integer, since no integer number raised to the power of 3 equals 3.
- Thus, 3 is not a perfect cube.
- For [tex]\(6x^3\)[/tex]:
- The coefficient is 6. The cube root of 6 is not an integer, since no integer number raised to the power of 3 equals 6.
- Thus, 6 is not a perfect cube.
- For [tex]\(9x^3\)[/tex]:
- The coefficient is 9. The cube root of 9 is not an integer, since no integer number raised to the power of 3 equals 9.
- Thus, 9 is not a perfect cube.
3. Identify the Perfect Cube:
- Out of these options, only [tex]\(1\)[/tex] is a perfect cube, because [tex]\(1^3 = 1\)[/tex].
Therefore, the monomial [tex]\(1x^3\)[/tex] is a perfect cube.
A monomial is a perfect cube when both the numerical coefficient and the variable's exponent are perfect cubes.
Here's how to solve it step-by-step:
1. Understand the Problem:
We have four monomials: [tex]\(1x^3\)[/tex], [tex]\(3x^3\)[/tex], [tex]\(6x^3\)[/tex], and [tex]\(9x^3\)[/tex]. We need to check if their coefficients are perfect cubes.
2. Check Each Coefficient:
- For [tex]\(1x^3\)[/tex]:
- The coefficient is 1. The cube root of 1 is [tex]\(1\)[/tex], since [tex]\(1^3 = 1\)[/tex].
- Thus, 1 is a perfect cube.
- For [tex]\(3x^3\)[/tex]:
- The coefficient is 3. The cube root of 3 is not an integer, since no integer number raised to the power of 3 equals 3.
- Thus, 3 is not a perfect cube.
- For [tex]\(6x^3\)[/tex]:
- The coefficient is 6. The cube root of 6 is not an integer, since no integer number raised to the power of 3 equals 6.
- Thus, 6 is not a perfect cube.
- For [tex]\(9x^3\)[/tex]:
- The coefficient is 9. The cube root of 9 is not an integer, since no integer number raised to the power of 3 equals 9.
- Thus, 9 is not a perfect cube.
3. Identify the Perfect Cube:
- Out of these options, only [tex]\(1\)[/tex] is a perfect cube, because [tex]\(1^3 = 1\)[/tex].
Therefore, the monomial [tex]\(1x^3\)[/tex] is a perfect cube.
