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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 analyze the monomial in terms of the properties of a perfect cube.
A perfect cube has the following properties:
1. The exponents of each variable in a perfect cube must be multiples of 3.
2. The coefficient (numerical part) must also be a perfect cube.
Let's break it down:
- Exponent of [tex]\(x\)[/tex]: The monomial has [tex]\(x^{18}\)[/tex]. Since 18 is a multiple of 3 (18 ÷ 3 = 6), the exponent of [tex]\(x\)[/tex] meets the requirement of being a perfect cube.
- Exponent of [tex]\(y\)[/tex]: The monomial has [tex]\(y^3\)[/tex]. Since 3 is a multiple of 3 (3 ÷ 3 = 1), this is already a perfect cube.
- Exponent of [tex]\(z\)[/tex]: The monomial has [tex]\(z^{21}\)[/tex]. Since 21 is a multiple of 3 (21 ÷ 3 = 7), this also satisfies the requirement for a perfect cube.
- Coefficient: The coefficient of the monomial is 215. For the monomial to be a perfect cube, this number must be a perfect cube too. However, 215 is not a perfect cube. The cube closest to 215 that would make it a perfect cube is 216, which is [tex]\(6^3\)[/tex].
Therefore, the number that needs to be changed in the monomial to make it a perfect cube is the coefficient 215. It should be replaced with 216.
A perfect cube has the following properties:
1. The exponents of each variable in a perfect cube must be multiples of 3.
2. The coefficient (numerical part) must also be a perfect cube.
Let's break it down:
- Exponent of [tex]\(x\)[/tex]: The monomial has [tex]\(x^{18}\)[/tex]. Since 18 is a multiple of 3 (18 ÷ 3 = 6), the exponent of [tex]\(x\)[/tex] meets the requirement of being a perfect cube.
- Exponent of [tex]\(y\)[/tex]: The monomial has [tex]\(y^3\)[/tex]. Since 3 is a multiple of 3 (3 ÷ 3 = 1), this is already a perfect cube.
- Exponent of [tex]\(z\)[/tex]: The monomial has [tex]\(z^{21}\)[/tex]. Since 21 is a multiple of 3 (21 ÷ 3 = 7), this also satisfies the requirement for a perfect cube.
- Coefficient: The coefficient of the monomial is 215. For the monomial to be a perfect cube, this number must be a perfect cube too. However, 215 is not a perfect cube. The cube closest to 215 that would make it a perfect cube is 216, which is [tex]\(6^3\)[/tex].
Therefore, the number that needs to be changed in the monomial to make it a perfect cube is the coefficient 215. It should be replaced with 216.
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