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Answer :
To make the monomial [tex]\(215x^{18}y^3z^{21}\)[/tex] a perfect cube, we need to ensure that the coefficient and all the exponents are multiples of 3.
Let's analyze each part of the monomial:
1. Coefficient (215):
- For the entire monomial to be a perfect cube, the coefficient itself must be a perfect cube. The number 215 is not currently a perfect cube. It can be broken down into its prime factors, which are 5 and 43. Neither of these can be rearranged into a cube, so the coefficient needs adjustment.
2. Exponents:
- [tex]\(x^{18}\)[/tex]: The exponent 18 is already a multiple of 3, so it's fine as is.
- [tex]\(y^3\)[/tex]: The exponent 3 is already a multiple of 3, so it's fine as is.
- [tex]\(z^{21}\)[/tex]: The exponent 21 is already a multiple of 3, so it's fine as is.
Since all the variable exponents are multiples of 3, they don't need any changes. So, the focus should be on the coefficient.
To make 215 a perfect cube, we find a number by which to multiply it so that the result is a cube. According to the question's solution, this missing factor is 46225. When 215 is multiplied by 46225, we get 9938375, which is a perfect cube.
In summary, the number in the monomial that needs to be changed for it to become a perfect cube is the coefficient, 215.
Let's analyze each part of the monomial:
1. Coefficient (215):
- For the entire monomial to be a perfect cube, the coefficient itself must be a perfect cube. The number 215 is not currently a perfect cube. It can be broken down into its prime factors, which are 5 and 43. Neither of these can be rearranged into a cube, so the coefficient needs adjustment.
2. Exponents:
- [tex]\(x^{18}\)[/tex]: The exponent 18 is already a multiple of 3, so it's fine as is.
- [tex]\(y^3\)[/tex]: The exponent 3 is already a multiple of 3, so it's fine as is.
- [tex]\(z^{21}\)[/tex]: The exponent 21 is already a multiple of 3, so it's fine as is.
Since all the variable exponents are multiples of 3, they don't need any changes. So, the focus should be on the coefficient.
To make 215 a perfect cube, we find a number by which to multiply it so that the result is a cube. According to the question's solution, this missing factor is 46225. When 215 is multiplied by 46225, we get 9938375, which is a perfect cube.
In summary, the number in the monomial that needs to be changed for it to become a perfect cube is the coefficient, 215.
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