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Answer :
To solve the problem of whether the monomial [tex]\(215x^{18}y^3z^{21}\)[/tex] is a perfect cube, we need to check both the exponents of the variables and the coefficient.
### Step 1: Check the Exponents
For a monomial to be a perfect cube, the exponents of all variables must be multiples of 3.
- Exponent of [tex]\(x\)[/tex]: The exponent is 18. Since [tex]\(18 \div 3 = 6\)[/tex], it is a multiple of 3.
- Exponent of [tex]\(y\)[/tex]: The exponent is 3. Since [tex]\(3 \div 3 = 1\)[/tex], it is a multiple of 3.
- Exponent of [tex]\(z\)[/tex]: The exponent is 21. Since [tex]\(21 \div 3 = 7\)[/tex], it is a multiple of 3.
All the exponents are already suitable for a perfect cube.
### Step 2: Check the Coefficient
For the coefficient to be part of a perfect cube, it must itself be a perfect cube.
- Coefficient 215: We need to determine if 215 is a perfect cube.
Since 215 is not a perfect cube, we should look for the smallest perfect cube that could replace it. The smallest perfect cube greater than 215 is 216, which is [tex]\(6^3\)[/tex].
### Conclusion
Therefore, to make the entire monomial a perfect cube, the number that needs to be changed is the coefficient 215, which should be changed to 216.
### Step 1: Check the Exponents
For a monomial to be a perfect cube, the exponents of all variables must be multiples of 3.
- Exponent of [tex]\(x\)[/tex]: The exponent is 18. Since [tex]\(18 \div 3 = 6\)[/tex], it is a multiple of 3.
- Exponent of [tex]\(y\)[/tex]: The exponent is 3. Since [tex]\(3 \div 3 = 1\)[/tex], it is a multiple of 3.
- Exponent of [tex]\(z\)[/tex]: The exponent is 21. Since [tex]\(21 \div 3 = 7\)[/tex], it is a multiple of 3.
All the exponents are already suitable for a perfect cube.
### Step 2: Check the Coefficient
For the coefficient to be part of a perfect cube, it must itself be a perfect cube.
- Coefficient 215: We need to determine if 215 is a perfect cube.
Since 215 is not a perfect cube, we should look for the smallest perfect cube that could replace it. The smallest perfect cube greater than 215 is 216, which is [tex]\(6^3\)[/tex].
### Conclusion
Therefore, to make the entire monomial a perfect cube, the number that needs to be changed is the coefficient 215, which should be changed to 216.
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