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 should understand the conditions for a monomial to be a perfect cube:
1. Perfect Cube Conditions:
- All exponents of the variables (or powers) must be divisible by 3.
- The coefficient (in this case, 215) must be a perfect cube.
2. Examine the Exponents:
- The exponent of [tex]\(x\)[/tex] is 18. Since 18 is divisible by 3, [tex]\(x^{18}\)[/tex] is already a perfect cube.
- The exponent of [tex]\(y\)[/tex] is 3. Since 3 is divisible by 3, [tex]\(y^3\)[/tex] is already a perfect cube.
- The exponent of [tex]\(z\)[/tex] is 21. Since 21 is divisible by 3, [tex]\(z^{21}\)[/tex] is already a perfect cube.
3. Examine the Coefficient:
- The coefficient is 215.
- For 215 to be a perfect cube, it would need to be equal to some integer raised to the power of 3.
- By examining cube numbers, the integer closest to the cube root of 215 is 6, because [tex]\(6^3 = 216\)[/tex].
- Therefore, 215 is not a perfect cube. To make the monomial a perfect cube, 215 should be changed to 216.
In conclusion, the number 215 in the monomial [tex]\(215 x^{18} y^3 z^{21}\)[/tex] needs to be changed to make the entire expression a perfect cube.
1. Perfect Cube Conditions:
- All exponents of the variables (or powers) must be divisible by 3.
- The coefficient (in this case, 215) must be a perfect cube.
2. Examine the Exponents:
- The exponent of [tex]\(x\)[/tex] is 18. Since 18 is divisible by 3, [tex]\(x^{18}\)[/tex] is already a perfect cube.
- The exponent of [tex]\(y\)[/tex] is 3. Since 3 is divisible by 3, [tex]\(y^3\)[/tex] is already a perfect cube.
- The exponent of [tex]\(z\)[/tex] is 21. Since 21 is divisible by 3, [tex]\(z^{21}\)[/tex] is already a perfect cube.
3. Examine the Coefficient:
- The coefficient is 215.
- For 215 to be a perfect cube, it would need to be equal to some integer raised to the power of 3.
- By examining cube numbers, the integer closest to the cube root of 215 is 6, because [tex]\(6^3 = 216\)[/tex].
- Therefore, 215 is not a perfect cube. To make the monomial a perfect cube, 215 should be changed to 216.
In conclusion, the number 215 in the monomial [tex]\(215 x^{18} y^3 z^{21}\)[/tex] needs to be changed to make the entire expression a perfect cube.
