Answer :
To determine which number in the monomial [tex]\(215 x^{18} y^3 z^{21}\)[/tex] needs to be changed to make it a perfect cube, let's go through each part of the monomial and analyze them:
1. Check the Coefficient: 215
- To be a perfect cube, a number's cube root must be an integer. The cube root of 215 is not an integer because 215 is not a perfect cube.
- A nearby perfect cube to 215 would be 216, since [tex]\(6^3 = 216\)[/tex]. Therefore, 215 should be changed to 216 to make it a perfect cube.
2. Check the Exponent of x: 18
- For the power of [tex]\(x\)[/tex] to be a perfect cube, the exponent must be divisible by 3. Since [tex]\(18 \div 3 = 6\)[/tex], the exponent 18 is already a multiple of 3, so it does not need to be changed.
3. Check the Exponent of y: 3
- The exponent for [tex]\(y\)[/tex] is 3, which is already a perfect cube since [tex]\(3 \div 3 = 1\)[/tex]. Therefore, it does not need any change.
4. Check the Exponent of z: 21
- Similarly, the exponent for [tex]\(z\)[/tex] is 21. Since [tex]\(21 \div 3 = 7\)[/tex], it is also already a multiple of 3, meaning the exponent 21 is fine as is.
From this analysis, we can see that the number which needs to be changed to turn the monomial into a perfect cube is 215. It should be changed to 216, as 216 is a perfect cube.
1. Check the Coefficient: 215
- To be a perfect cube, a number's cube root must be an integer. The cube root of 215 is not an integer because 215 is not a perfect cube.
- A nearby perfect cube to 215 would be 216, since [tex]\(6^3 = 216\)[/tex]. Therefore, 215 should be changed to 216 to make it a perfect cube.
2. Check the Exponent of x: 18
- For the power of [tex]\(x\)[/tex] to be a perfect cube, the exponent must be divisible by 3. Since [tex]\(18 \div 3 = 6\)[/tex], the exponent 18 is already a multiple of 3, so it does not need to be changed.
3. Check the Exponent of y: 3
- The exponent for [tex]\(y\)[/tex] is 3, which is already a perfect cube since [tex]\(3 \div 3 = 1\)[/tex]. Therefore, it does not need any change.
4. Check the Exponent of z: 21
- Similarly, the exponent for [tex]\(z\)[/tex] is 21. Since [tex]\(21 \div 3 = 7\)[/tex], it is also already a multiple of 3, meaning the exponent 21 is fine as is.
From this analysis, we can see that the number which needs to be changed to turn the monomial into a perfect cube is 215. It should be changed to 216, as 216 is a perfect cube.
