Answer :
To determine which number in the monomial [tex]\(215 x^{18} y^3 z^{21}\)[/tex] needs to be changed for it to be a perfect cube, let's break it down step by step:
1. Understanding a Perfect Cube:
For a monomial to be a perfect cube, all the variable exponents must be multiples of 3, and the coefficient (the numerical part) should be a perfect cube itself.
2. Analyze the Exponents:
- The exponent of [tex]\(x\)[/tex] is 18. Since 18 is divisible by 3 (18 ÷ 3 = 6), this exponent is fine for a perfect cube.
- The exponent of [tex]\(y\)[/tex] is 3. Since 3 is divisible by 3 (3 ÷ 3 = 1), this exponent is also fine for a perfect cube.
- The exponent of [tex]\(z\)[/tex] is 21. Since 21 is divisible by 3 (21 ÷ 3 = 7), this exponent is suitable for a perfect cube as well.
3. Analyze the Coefficient:
- The coefficient is 215. For it to be part of a perfect cube monomial, 215 itself should be a perfect cube, meaning it should be expressible as [tex]\(n^3\)[/tex] for some integer [tex]\(n\)[/tex].
- Checking 215, there is no integer that, when cubed, results in 215. Therefore, 215 is not a perfect cube.
4. Conclusion:
Since the exponents of the variables [tex]\(x\)[/tex], [tex]\(y\)[/tex], and [tex]\(z\)[/tex] are already suitable for forming a perfect cube and the number that needs to be changed is the coefficient, the number that must be adjusted is 215.
Therefore, to make the monomial [tex]\(215 x^{18} y^3 z^{21}\)[/tex] a perfect cube, the number 215 needs to be changed.
1. Understanding a Perfect Cube:
For a monomial to be a perfect cube, all the variable exponents must be multiples of 3, and the coefficient (the numerical part) should be a perfect cube itself.
2. Analyze the Exponents:
- The exponent of [tex]\(x\)[/tex] is 18. Since 18 is divisible by 3 (18 ÷ 3 = 6), this exponent is fine for a perfect cube.
- The exponent of [tex]\(y\)[/tex] is 3. Since 3 is divisible by 3 (3 ÷ 3 = 1), this exponent is also fine for a perfect cube.
- The exponent of [tex]\(z\)[/tex] is 21. Since 21 is divisible by 3 (21 ÷ 3 = 7), this exponent is suitable for a perfect cube as well.
3. Analyze the Coefficient:
- The coefficient is 215. For it to be part of a perfect cube monomial, 215 itself should be a perfect cube, meaning it should be expressible as [tex]\(n^3\)[/tex] for some integer [tex]\(n\)[/tex].
- Checking 215, there is no integer that, when cubed, results in 215. Therefore, 215 is not a perfect cube.
4. Conclusion:
Since the exponents of the variables [tex]\(x\)[/tex], [tex]\(y\)[/tex], and [tex]\(z\)[/tex] are already suitable for forming a perfect cube and the number that needs to be changed is the coefficient, the number that must be adjusted is 215.
Therefore, to make the monomial [tex]\(215 x^{18} y^3 z^{21}\)[/tex] a perfect cube, the number 215 needs to be changed.
