Answer :
To determine which part of the monomial [tex]\(215x^{18}y^3z^{21}\)[/tex] needs to be changed to make it a perfect cube, let's break down the problem:
1. Understanding Perfect Cubes:
- A number or term is a perfect cube if it can be expressed as another number or term raised to the power of 3. Essentially, everything in the expression must be divisible by 3.
2. Analyzing the Monomial:
- The monomial is [tex]\(215x^{18}y^3z^{21}\)[/tex].
- We need to verify if each part (the coefficient and the variables' exponents) is already a perfect cube.
3. Checking the Exponents:
- The exponent of [tex]\(x\)[/tex] is 18. Since 18 is divisible by 3 (18 ÷ 3 = 6), [tex]\(x^{18}\)[/tex] is already a perfect cube.
- The exponent of [tex]\(y\)[/tex] is 3. Since 3 is divisible by 3 (3 ÷ 3 = 1), [tex]\(y^3\)[/tex] is already a perfect cube.
- The exponent of [tex]\(z\)[/tex] is 21. Since 21 is divisible by 3 (21 ÷ 3 = 7), [tex]\(z^{21}\)[/tex] is already a perfect cube.
4. Checking the Coefficient:
- Now, we look at the coefficient 215. For the whole monomial to be a perfect cube, 215 itself must also be a perfect cube.
- A perfect cube coefficient implies that the cube root of 215 should be an integer.
- Since 215 is not a perfect cube (as it doesn't equal any integer cubed), the coefficient 215 must be changed to make the entire monomial a perfect cube.
In conclusion, the number in the monomial that needs to be changed to make it a perfect cube is [tex]\(215\)[/tex].
1. Understanding Perfect Cubes:
- A number or term is a perfect cube if it can be expressed as another number or term raised to the power of 3. Essentially, everything in the expression must be divisible by 3.
2. Analyzing the Monomial:
- The monomial is [tex]\(215x^{18}y^3z^{21}\)[/tex].
- We need to verify if each part (the coefficient and the variables' exponents) is already a perfect cube.
3. Checking the Exponents:
- The exponent of [tex]\(x\)[/tex] is 18. Since 18 is divisible by 3 (18 ÷ 3 = 6), [tex]\(x^{18}\)[/tex] is already a perfect cube.
- The exponent of [tex]\(y\)[/tex] is 3. Since 3 is divisible by 3 (3 ÷ 3 = 1), [tex]\(y^3\)[/tex] is already a perfect cube.
- The exponent of [tex]\(z\)[/tex] is 21. Since 21 is divisible by 3 (21 ÷ 3 = 7), [tex]\(z^{21}\)[/tex] is already a perfect cube.
4. Checking the Coefficient:
- Now, we look at the coefficient 215. For the whole monomial to be a perfect cube, 215 itself must also be a perfect cube.
- A perfect cube coefficient implies that the cube root of 215 should be an integer.
- Since 215 is not a perfect cube (as it doesn't equal any integer cubed), the coefficient 215 must be changed to make the entire monomial a perfect cube.
In conclusion, the number in the monomial that needs to be changed to make it a perfect cube is [tex]\(215\)[/tex].
