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 must ensure that all components of the expression are perfect cubes.
1. Check the Coefficient:
- The coefficient is 215.
- A number is a perfect cube if it can be written as [tex]\(n^3\)[/tex], where [tex]\(n\)[/tex] is an integer.
- To be a perfect cube, 215 would need to be decomposed into prime factors, but it needs no calculations here.
- 215 is not a perfect cube because its factors cannot be organized into groups of three equal parts.
2. Check the Exponents:
- For [tex]\(x\)[/tex]: The exponent is 18.
- A number is a perfect cube if the exponent is a multiple of 3.
- 18 is a multiple of 3 [tex]\((18 \div 3 = 6)\)[/tex].
- For [tex]\(y\)[/tex]: The exponent is 3.
- 3 is already a multiple of 3 [tex]\((3 \div 3 = 1)\)[/tex].
- For [tex]\(z\)[/tex]: The exponent is 21.
- 21 is a multiple of 3 [tex]\((21 \div 3 = 7)\)[/tex].
Since all the variables already have exponents that are multiples of 3, the only concern is the coefficient 215.
To convert the entire expression into a perfect cube, the coefficient 215 needs to be changed to a number that is a perfect cube. Thus, the number that needs to be changed is 215.
1. Check the Coefficient:
- The coefficient is 215.
- A number is a perfect cube if it can be written as [tex]\(n^3\)[/tex], where [tex]\(n\)[/tex] is an integer.
- To be a perfect cube, 215 would need to be decomposed into prime factors, but it needs no calculations here.
- 215 is not a perfect cube because its factors cannot be organized into groups of three equal parts.
2. Check the Exponents:
- For [tex]\(x\)[/tex]: The exponent is 18.
- A number is a perfect cube if the exponent is a multiple of 3.
- 18 is a multiple of 3 [tex]\((18 \div 3 = 6)\)[/tex].
- For [tex]\(y\)[/tex]: The exponent is 3.
- 3 is already a multiple of 3 [tex]\((3 \div 3 = 1)\)[/tex].
- For [tex]\(z\)[/tex]: The exponent is 21.
- 21 is a multiple of 3 [tex]\((21 \div 3 = 7)\)[/tex].
Since all the variables already have exponents that are multiples of 3, the only concern is the coefficient 215.
To convert the entire expression into a perfect cube, the coefficient 215 needs to be changed to a number that is a perfect cube. Thus, the number that needs to be changed is 215.
