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Answer :
To determine which monomial is a perfect cube among the given options, let's analyze each monomial separately.
The options are:
1. [tex]\( x^3 \)[/tex]
2. [tex]\( 3x^3 \)[/tex]
3. [tex]\( 6x^3 \)[/tex]
4. [tex]\( 9x^3 \)[/tex]
When dealing with perfect cubes, both the coefficient (the number in front of the variable) and the variable's exponent should be checked.
1. Monomial: [tex]\( x^3 \)[/tex]
- Coefficient: 1
- The number 1 is already a perfect cube because [tex]\( 1^3 = 1 \)[/tex].
- The exponent of [tex]\( x \)[/tex] is 3, which means [tex]\( (x^3) \)[/tex] itself is a perfect cube.
2. Monomial: [tex]\( 3x^3 \)[/tex]
- Coefficient: 3
- The number 3 is not a perfect cube because there is no integer that, when cubed, equals 3.
3. Monomial: [tex]\( 6x^3 \)[/tex]
- Coefficient: 6
- The number 6 is not a perfect cube because there is no integer that, when cubed, equals 6.
4. Monomial: [tex]\( 9x^3 \)[/tex]
- Coefficient: 9
- The number 9 is not a perfect cube because there is no integer that, when cubed, equals 9.
Based on this analysis, the only monomial among the given options that is a perfect cube is [tex]\( x^3 \)[/tex], as it is both mathematically correct for the coefficient (1, which is a perfect cube), and for the variable part because the exponent is 3.
The options are:
1. [tex]\( x^3 \)[/tex]
2. [tex]\( 3x^3 \)[/tex]
3. [tex]\( 6x^3 \)[/tex]
4. [tex]\( 9x^3 \)[/tex]
When dealing with perfect cubes, both the coefficient (the number in front of the variable) and the variable's exponent should be checked.
1. Monomial: [tex]\( x^3 \)[/tex]
- Coefficient: 1
- The number 1 is already a perfect cube because [tex]\( 1^3 = 1 \)[/tex].
- The exponent of [tex]\( x \)[/tex] is 3, which means [tex]\( (x^3) \)[/tex] itself is a perfect cube.
2. Monomial: [tex]\( 3x^3 \)[/tex]
- Coefficient: 3
- The number 3 is not a perfect cube because there is no integer that, when cubed, equals 3.
3. Monomial: [tex]\( 6x^3 \)[/tex]
- Coefficient: 6
- The number 6 is not a perfect cube because there is no integer that, when cubed, equals 6.
4. Monomial: [tex]\( 9x^3 \)[/tex]
- Coefficient: 9
- The number 9 is not a perfect cube because there is no integer that, when cubed, equals 9.
Based on this analysis, the only monomial among the given options that is a perfect cube is [tex]\( x^3 \)[/tex], as it is both mathematically correct for the coefficient (1, which is a perfect cube), and for the variable part because the exponent is 3.
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