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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, we need to check each part of the monomial: the coefficient and the exponents. A monomial is a perfect cube if every part of it is a perfect cube or can be expressed in terms of cubes.
1. Coefficients:
- The coefficient is 215. For the monomial to be a perfect cube, 215 itself must be a perfect cube, which means there should be some integer [tex]\(a\)[/tex] such that [tex]\(a^3 = 215\)[/tex].
- Upon checking, [tex]\( \sqrt[3]{215} \)[/tex] does not yield an integer. This implies that 215 is not a perfect cube.
2. Exponents:
- For the monomial to be a perfect cube, the exponents of each variable must be multiples of 3.
- The exponent of [tex]\(x\)[/tex] is 18. Since [tex]\(18 \div 3 = 6\)[/tex], it is already a multiple of 3.
- The exponent of [tex]\(y\)[/tex] is 3. Since [tex]\(3 \div 3 = 1\)[/tex], it is also a multiple of 3.
- The exponent of [tex]\(z\)[/tex] is 21. Since [tex]\(21 \div 3 = 7\)[/tex], it is a multiple of 3 as well.
Given the analysis above, we find that the only component that is not a perfect cube or that doesn't fit is the number 215. Therefore, 215 needs to be changed to make the entire monomial a perfect cube.
1. Coefficients:
- The coefficient is 215. For the monomial to be a perfect cube, 215 itself must be a perfect cube, which means there should be some integer [tex]\(a\)[/tex] such that [tex]\(a^3 = 215\)[/tex].
- Upon checking, [tex]\( \sqrt[3]{215} \)[/tex] does not yield an integer. This implies that 215 is not a perfect cube.
2. Exponents:
- For the monomial to be a perfect cube, the exponents of each variable must be multiples of 3.
- The exponent of [tex]\(x\)[/tex] is 18. Since [tex]\(18 \div 3 = 6\)[/tex], it is already a multiple of 3.
- The exponent of [tex]\(y\)[/tex] is 3. Since [tex]\(3 \div 3 = 1\)[/tex], it is also a multiple of 3.
- The exponent of [tex]\(z\)[/tex] is 21. Since [tex]\(21 \div 3 = 7\)[/tex], it is a multiple of 3 as well.
Given the analysis above, we find that the only component that is not a perfect cube or that doesn't fit is the number 215. Therefore, 215 needs to be changed to make the entire monomial a perfect cube.
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