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 examine each part separately.
1. Examine the exponents:
- For a monomial to be a perfect cube, each exponent must be a multiple of 3.
- The exponent of [tex]\(x\)[/tex] is 18, which is a multiple of 3.
- The exponent of [tex]\(y\)[/tex] is 3, which is a multiple of 3.
- The exponent of [tex]\(z\)[/tex] is 21, which is a multiple of 3.
So, the exponents of all the variables already satisfy the condition for being part of a perfect cube.
2. Examine the coefficient 215:
- We need to determine if 215 is a perfect cube. A perfect cube is a number that can be expressed as [tex]\(n^3\)[/tex], where [tex]\(n\)[/tex] is an integer.
- The cube root of 215 is approximately 5.99.
Let's explore numbers around this value:
- The cube of 5 is [tex]\(5^3 = 125\)[/tex].
- The cube of 6 is [tex]\(6^3 = 216\)[/tex].
215 is not a perfect cube because it does not equal either 125 or 216 (the nearest perfect cubes). Therefore, the coefficient 215 should be changed.
Conclusion:
To make the monomial a perfect cube, 215 should be adjusted to a perfect cube, such as 216. Among the given options, the number that needs to be changed is 215.
1. Examine the exponents:
- For a monomial to be a perfect cube, each exponent must be a multiple of 3.
- The exponent of [tex]\(x\)[/tex] is 18, which is a multiple of 3.
- The exponent of [tex]\(y\)[/tex] is 3, which is a multiple of 3.
- The exponent of [tex]\(z\)[/tex] is 21, which is a multiple of 3.
So, the exponents of all the variables already satisfy the condition for being part of a perfect cube.
2. Examine the coefficient 215:
- We need to determine if 215 is a perfect cube. A perfect cube is a number that can be expressed as [tex]\(n^3\)[/tex], where [tex]\(n\)[/tex] is an integer.
- The cube root of 215 is approximately 5.99.
Let's explore numbers around this value:
- The cube of 5 is [tex]\(5^3 = 125\)[/tex].
- The cube of 6 is [tex]\(6^3 = 216\)[/tex].
215 is not a perfect cube because it does not equal either 125 or 216 (the nearest perfect cubes). Therefore, the coefficient 215 should be changed.
Conclusion:
To make the monomial a perfect cube, 215 should be adjusted to a perfect cube, such as 216. Among the given options, the number that needs to be changed is 215.
