Answer :
Sure, let's break this problem down step by step.
### Step-by-Step Solution:
1. Understanding the Problem:
We need to determine which monomial among the given choices is a perfect cube.
2. Identify the Choices:
We have four monomials to consider:
[tex]\[
1x^3, \quad 3x^3, \quad 6x^3, \quad 9x^3
\][/tex]
Here, we observe that each monomial has the variable part [tex]\( x^3 \)[/tex], which is already a perfect cube. So, we just need to check the coefficients.
3. Check the Coefficients:
We need to identify if the coefficients (the numbers in front of [tex]\( x^3 \)[/tex]) are perfect cubes. The coefficients given are:
[tex]\[
1, \quad 3, \quad 6, \quad 9
\][/tex]
4. Determine if Each Coefficient is a Perfect Cube:
- 1: A perfect cube of 1 is [tex]\( 1 \times 1 \times 1 = 1 \)[/tex]. So, 1 is a perfect cube.
- 3: The cube root of 3 is approximately [tex]\( 1.442 \)[/tex]. [tex]\( 1.442 \times 1.442 \times 1.442 \)[/tex] does not equal 3. So, 3 is not a perfect cube.
- 6: The cube root of 6 is approximately [tex]\( 1.817 \)[/tex]. [tex]\( 1.817 \times 1.817 \times 1.817 \)[/tex] does not equal 6. So, 6 is not a perfect cube.
- 9: The cube root of 9 is approximately [tex]\( 2.080 \)[/tex]. [tex]\( 2.080 \times 2.080 \times 2.080 \)[/tex] does not equal 9. So, 9 is not a perfect cube.
5. Conclusion:
Among the given coefficients, only 1 is a perfect cube.
So, the monomial that is a perfect cube is:
[tex]\[
1x^3
\][/tex]
### Step-by-Step Solution:
1. Understanding the Problem:
We need to determine which monomial among the given choices is a perfect cube.
2. Identify the Choices:
We have four monomials to consider:
[tex]\[
1x^3, \quad 3x^3, \quad 6x^3, \quad 9x^3
\][/tex]
Here, we observe that each monomial has the variable part [tex]\( x^3 \)[/tex], which is already a perfect cube. So, we just need to check the coefficients.
3. Check the Coefficients:
We need to identify if the coefficients (the numbers in front of [tex]\( x^3 \)[/tex]) are perfect cubes. The coefficients given are:
[tex]\[
1, \quad 3, \quad 6, \quad 9
\][/tex]
4. Determine if Each Coefficient is a Perfect Cube:
- 1: A perfect cube of 1 is [tex]\( 1 \times 1 \times 1 = 1 \)[/tex]. So, 1 is a perfect cube.
- 3: The cube root of 3 is approximately [tex]\( 1.442 \)[/tex]. [tex]\( 1.442 \times 1.442 \times 1.442 \)[/tex] does not equal 3. So, 3 is not a perfect cube.
- 6: The cube root of 6 is approximately [tex]\( 1.817 \)[/tex]. [tex]\( 1.817 \times 1.817 \times 1.817 \)[/tex] does not equal 6. So, 6 is not a perfect cube.
- 9: The cube root of 9 is approximately [tex]\( 2.080 \)[/tex]. [tex]\( 2.080 \times 2.080 \times 2.080 \)[/tex] does not equal 9. So, 9 is not a perfect cube.
5. Conclusion:
Among the given coefficients, only 1 is a perfect cube.
So, the monomial that is a perfect cube is:
[tex]\[
1x^3
\][/tex]
