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 check that each part of the monomial is a perfect cube itself.
For a monomial to be a perfect cube:
1. The coefficient (the number in front) should be a perfect cube.
2. Each exponent in the variables should be a multiple of 3.
Let's analyze each part:
1. Coefficient: 215
To be a perfect cube, the number needs to be something raised to the third power (for example, [tex]\(1^3 = 1\)[/tex], [tex]\(2^3 = 8\)[/tex], [tex]\(3^3 = 27\)[/tex], etc.).
Let's factorize 215:
[tex]\[
215 = 5 \times 43
\][/tex]
Since both 5 and 43 are prime numbers and not perfect cubes themselves, 215 is not a perfect cube. Therefore, the coefficient 215 needs to change to make the monomial a perfect cube.
2. Exponent of x: 18
Since 18 is divisible by 3 ([tex]\(18 \div 3 = 6\)[/tex]), the exponent of [tex]\(x^{18}\)[/tex] is already a multiple of 3, so it's fine.
3. Exponent of y: 3
Since 3 is divisible by 3 ([tex]\(3 \div 3 = 1\)[/tex]), the exponent of [tex]\(y^3\)[/tex] is fine.
4. Exponent of z: 21
Since 21 is divisible by 3 ([tex]\(21 \div 3 = 7\)[/tex]), the exponent of [tex]\(z^{21}\)[/tex] is fine.
In conclusion, to make the whole monomial a perfect cube, only the coefficient 215 needs to be changed. It needs to be converted to a perfect cube, like 216 ([tex]\(6^3 = 216\)[/tex]). Thus, the number in the monomial that needs to change is:
215
For a monomial to be a perfect cube:
1. The coefficient (the number in front) should be a perfect cube.
2. Each exponent in the variables should be a multiple of 3.
Let's analyze each part:
1. Coefficient: 215
To be a perfect cube, the number needs to be something raised to the third power (for example, [tex]\(1^3 = 1\)[/tex], [tex]\(2^3 = 8\)[/tex], [tex]\(3^3 = 27\)[/tex], etc.).
Let's factorize 215:
[tex]\[
215 = 5 \times 43
\][/tex]
Since both 5 and 43 are prime numbers and not perfect cubes themselves, 215 is not a perfect cube. Therefore, the coefficient 215 needs to change to make the monomial a perfect cube.
2. Exponent of x: 18
Since 18 is divisible by 3 ([tex]\(18 \div 3 = 6\)[/tex]), the exponent of [tex]\(x^{18}\)[/tex] is already a multiple of 3, so it's fine.
3. Exponent of y: 3
Since 3 is divisible by 3 ([tex]\(3 \div 3 = 1\)[/tex]), the exponent of [tex]\(y^3\)[/tex] is fine.
4. Exponent of z: 21
Since 21 is divisible by 3 ([tex]\(21 \div 3 = 7\)[/tex]), the exponent of [tex]\(z^{21}\)[/tex] is fine.
In conclusion, to make the whole monomial a perfect cube, only the coefficient 215 needs to be changed. It needs to be converted to a perfect cube, like 216 ([tex]\(6^3 = 216\)[/tex]). Thus, the number in the monomial that needs to change is:
215
