To determine which monomial is a perfect cube, we need to examine each monomial to see if both the coefficient and the exponent are perfect cubes.
Let's break it down step-by-step:
### Step 1: Check the Coefficients
We need to see if the coefficient of each monomial is a perfect cube.
1. Coefficient = 1
- [tex]\(1 = 1^3\)[/tex] (True, 1 is a perfect cube)
2. Coefficient = 8
- [tex]\(8 = 2^3\)[/tex] (True, 8 is a perfect cube)
3. Coefficient = 9
- [tex]\(9 = 2.08008^3\)[/tex] (Approximately, but not exact. False, 9 is not a perfect cube)
4. Coefficient = 27
- [tex]\(27 = 3^3\)[/tex] (True, 27 is a perfect cube)
### Step 2: Check the Exponents
Next, we need to check if the exponent of the variable x in each monomial is divisible by 3.
1. Exponent = 10
- [tex]\(10 \mod 3 = 1\)[/tex] (False, 10 is not divisible by 3)
2. Exponent = 8
- [tex]\(8 \mod 3 = 2\)[/tex] (False, 8 is not divisible by 3)
3. Exponent = 9
- [tex]\(9 \mod 3 = 0\)[/tex] (True, 9 is divisible by 3)
4. Exponent = 15
- [tex]\(15 \mod 3 = 0\)[/tex] (True, 15 is divisible by 3)
### Step 3: Combine Checks for Coefficients and Exponents
Combine the results of the coefficient and exponent checks:
1. [tex]\(1 x^{10}\)[/tex]
- Coefficient = True (1)
- Exponent = False (10)
- Combined = False (Does not meet both conditions)
2. [tex]\(8 x^8\)[/tex]
- Coefficient = True (8)
- Exponent = False (8)
- Combined = False (Does not meet both conditions)
3. [tex]\(9 x^9\)[/tex]
- Coefficient = False (9)
- Exponent = True (9)
- Combined = False (Does not meet both conditions)
4. [tex]\(27 x^{15}\)[/tex]
- Coefficient = True (27)
- Exponent = True (15)
- Combined = True (Meets both conditions)
### Conclusion
The monomial [tex]\(27 x^{15}\)[/tex] is a perfect cube because both its coefficient (27) is a perfect cube and its exponent (15) is divisible by 3.
Thus, the perfect cube among the given monomials is [tex]\(27 x^{15}\)[/tex].
