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 understand the components required for something to be a perfect cube:
1. Definition of a Perfect Cube: A perfect cube is a number or expression that can be expressed as some integer or term raised to the power of 3.
2. Monomial Perfect Cube Requirements:
- The constant term (number) should be a perfect cube.
- The exponents of all variables should be multiples of 3.
Now, we evaluate each component of the monomial:
- Constant Term (215): For 215 to be a perfect cube, it needs to be expressible as [tex]\(n^3\)[/tex], where [tex]\(n\)[/tex] is an integer. The prime factorization of 215 is [tex]\(5 \times 43\)[/tex]. Since neither 5 nor 43 is a perfect cube, 215 is not a perfect cube.
- Exponent of [tex]\(x\)[/tex] (18): Since 18 is divisible by 3 (18 ÷ 3 = 6), the exponent for [tex]\(x\)[/tex] fits the criteria for a perfect cube.
- Exponent of [tex]\(y\)[/tex] (3): Since 3 is divisible by 3 (3 ÷ 3 = 1), the exponent for [tex]\(y\)[/tex] also fits the criteria for a perfect cube.
- Exponent of [tex]\(z\)[/tex] (21): Since 21 is divisible by 3 (21 ÷ 3 = 7), the exponent for [tex]\(z\)[/tex] fits the criteria for a perfect cube as well.
Based on these checks, the component that needs to be changed to make the entire expression a perfect cube is the constant term, 215, since it is not a perfect cube.
Therefore, the number in the monomial that needs to be changed is 215.
1. Definition of a Perfect Cube: A perfect cube is a number or expression that can be expressed as some integer or term raised to the power of 3.
2. Monomial Perfect Cube Requirements:
- The constant term (number) should be a perfect cube.
- The exponents of all variables should be multiples of 3.
Now, we evaluate each component of the monomial:
- Constant Term (215): For 215 to be a perfect cube, it needs to be expressible as [tex]\(n^3\)[/tex], where [tex]\(n\)[/tex] is an integer. The prime factorization of 215 is [tex]\(5 \times 43\)[/tex]. Since neither 5 nor 43 is a perfect cube, 215 is not a perfect cube.
- Exponent of [tex]\(x\)[/tex] (18): Since 18 is divisible by 3 (18 ÷ 3 = 6), the exponent for [tex]\(x\)[/tex] fits the criteria for a perfect cube.
- Exponent of [tex]\(y\)[/tex] (3): Since 3 is divisible by 3 (3 ÷ 3 = 1), the exponent for [tex]\(y\)[/tex] also fits the criteria for a perfect cube.
- Exponent of [tex]\(z\)[/tex] (21): Since 21 is divisible by 3 (21 ÷ 3 = 7), the exponent for [tex]\(z\)[/tex] fits the criteria for a perfect cube as well.
Based on these checks, the component that needs to be changed to make the entire expression a perfect cube is the constant term, 215, since it is not a perfect cube.
Therefore, the number in the monomial that needs to be changed is 215.
