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To determine which of the given expressions is a perfect cube, we need to check whether the coefficients and the powers of the variables [tex]\(x\)[/tex] and [tex]\(y\)[/tex] are all perfect cubes.
A perfect cube for a number [tex]\( n \)[/tex] is a number that can be expressed as [tex]\( n = k^3 \)[/tex] where [tex]\( k \)[/tex] is an integer. Similarly, for an expression [tex]\( x^a y^b \)[/tex], it is a perfect cube if both [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are divisible by 3.
Let's analyze each expression step-by-step:
1. Expression: [tex]\(-8 x^{21} y^8\)[/tex]
- Coefficient: [tex]\(-8\)[/tex]
- [tex]\( -8 = (-2)^3 \)[/tex] (Yes, [tex]\(-8\)[/tex] is a perfect cube)
- Power of [tex]\( x \)[/tex]: [tex]\( 21 \)[/tex]
- [tex]\( 21 \div 3 = 7 \)[/tex] (Yes, 21 is divisible by 3)
- Power of [tex]\( y \)[/tex]: [tex]\( 8 \)[/tex]
- [tex]\( 8 \div 3 \approx 2.67 \)[/tex] (No, 8 is not divisible by 3)
- Therefore, [tex]\(-8 x^{21} y^8\)[/tex] is not a perfect cube.
2. Expression: [tex]\(-64 x^{64} y^{64}\)[/tex]
- Coefficient: [tex]\(-64\)[/tex]
- [tex]\( -64 = (-4)^3 \)[/tex] (No, [tex]\(-64 = (-4)^3\)[/tex] is not correct as [tex]\((-4)^3 = -64\)[/tex])
- Therefore, [tex]\(-64 x^{64} y^{64}\)[/tex] is not a perfect cube.
3. Expression: [tex]\(-125 x^8 y^{20}\)[/tex]
- Coefficient: [tex]\(-125\)[/tex]
- [tex]\( -125 = (-5)^3 \)[/tex] (Yes, [tex]\(-125\)[/tex] is a perfect cube)
- Power of [tex]\( x \)[/tex]: [tex]\( 8 \)[/tex]
- [tex]\( 8 \div 3 \approx 2.67 \)[/tex] (No, 8 is not divisible by 3)
- Power of [tex]\( y \)[/tex]: [tex]\( 20 \)[/tex]
- [tex]\( 20 \div 3 \approx 6.67 \)[/tex] (No, 20 is not divisible by 3)
- Therefore, [tex]\(-125 x^8 y^{20}\)[/tex] is not a perfect cube.
4. Expression: [tex]\(-216 x^8 y^{18}\)[/tex]
- Coefficient: [tex]\(-216\)[/tex]
- [tex]\(-216 = (-6)^3 \)[/tex] (Yes, [tex]\(-216\)[/tex] is a perfect cube)
- Power of [tex]\( x \)[/tex]: [tex]\( 8 \)[/tex]
- [tex]\( 8 \div 3 \approx 2.67 \)[/tex] (No, 8 is not divisible by 3)
- Power of [tex]\( y \)[/tex]: [tex]\( 18 \)[/tex]
- [tex]\( 18 \div 3 = 6 \)[/tex] (Yes, 18 is divisible by 3)
- Therefore, [tex]\(-216 x^8 y^{18}\)[/tex] is not a perfect cube.
After checking each expression:
- [tex]\(-8 x^{21} y^8\)[/tex] is not a perfect cube because [tex]\( y^8 \)[/tex] is not a perfect cube.
- [tex]\(-64 x^{64} y^{64}\)[/tex] is not a perfect cube because [tex]\(-64\)[/tex] is a perfect cube but the coefficient and its cubes do not justify to check the powers.
- [tex]\(-125 x^8 y^{20}\)[/tex] is not a perfect cube because neither [tex]\(x^8\)[/tex] nor [tex]\(y^{20}\)[/tex] are divisible by 3.
- [tex]\(-216 x^8 y^{18}\)[/tex] is not a perfect cube because [tex]\(x^8\)[/tex] is not divisible by 3.
None of the given expressions are perfect cubes.
A perfect cube for a number [tex]\( n \)[/tex] is a number that can be expressed as [tex]\( n = k^3 \)[/tex] where [tex]\( k \)[/tex] is an integer. Similarly, for an expression [tex]\( x^a y^b \)[/tex], it is a perfect cube if both [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are divisible by 3.
Let's analyze each expression step-by-step:
1. Expression: [tex]\(-8 x^{21} y^8\)[/tex]
- Coefficient: [tex]\(-8\)[/tex]
- [tex]\( -8 = (-2)^3 \)[/tex] (Yes, [tex]\(-8\)[/tex] is a perfect cube)
- Power of [tex]\( x \)[/tex]: [tex]\( 21 \)[/tex]
- [tex]\( 21 \div 3 = 7 \)[/tex] (Yes, 21 is divisible by 3)
- Power of [tex]\( y \)[/tex]: [tex]\( 8 \)[/tex]
- [tex]\( 8 \div 3 \approx 2.67 \)[/tex] (No, 8 is not divisible by 3)
- Therefore, [tex]\(-8 x^{21} y^8\)[/tex] is not a perfect cube.
2. Expression: [tex]\(-64 x^{64} y^{64}\)[/tex]
- Coefficient: [tex]\(-64\)[/tex]
- [tex]\( -64 = (-4)^3 \)[/tex] (No, [tex]\(-64 = (-4)^3\)[/tex] is not correct as [tex]\((-4)^3 = -64\)[/tex])
- Therefore, [tex]\(-64 x^{64} y^{64}\)[/tex] is not a perfect cube.
3. Expression: [tex]\(-125 x^8 y^{20}\)[/tex]
- Coefficient: [tex]\(-125\)[/tex]
- [tex]\( -125 = (-5)^3 \)[/tex] (Yes, [tex]\(-125\)[/tex] is a perfect cube)
- Power of [tex]\( x \)[/tex]: [tex]\( 8 \)[/tex]
- [tex]\( 8 \div 3 \approx 2.67 \)[/tex] (No, 8 is not divisible by 3)
- Power of [tex]\( y \)[/tex]: [tex]\( 20 \)[/tex]
- [tex]\( 20 \div 3 \approx 6.67 \)[/tex] (No, 20 is not divisible by 3)
- Therefore, [tex]\(-125 x^8 y^{20}\)[/tex] is not a perfect cube.
4. Expression: [tex]\(-216 x^8 y^{18}\)[/tex]
- Coefficient: [tex]\(-216\)[/tex]
- [tex]\(-216 = (-6)^3 \)[/tex] (Yes, [tex]\(-216\)[/tex] is a perfect cube)
- Power of [tex]\( x \)[/tex]: [tex]\( 8 \)[/tex]
- [tex]\( 8 \div 3 \approx 2.67 \)[/tex] (No, 8 is not divisible by 3)
- Power of [tex]\( y \)[/tex]: [tex]\( 18 \)[/tex]
- [tex]\( 18 \div 3 = 6 \)[/tex] (Yes, 18 is divisible by 3)
- Therefore, [tex]\(-216 x^8 y^{18}\)[/tex] is not a perfect cube.
After checking each expression:
- [tex]\(-8 x^{21} y^8\)[/tex] is not a perfect cube because [tex]\( y^8 \)[/tex] is not a perfect cube.
- [tex]\(-64 x^{64} y^{64}\)[/tex] is not a perfect cube because [tex]\(-64\)[/tex] is a perfect cube but the coefficient and its cubes do not justify to check the powers.
- [tex]\(-125 x^8 y^{20}\)[/tex] is not a perfect cube because neither [tex]\(x^8\)[/tex] nor [tex]\(y^{20}\)[/tex] are divisible by 3.
- [tex]\(-216 x^8 y^{18}\)[/tex] is not a perfect cube because [tex]\(x^8\)[/tex] is not divisible by 3.
None of the given expressions are perfect cubes.
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