Answer :
To determine which expressions are perfect cubes, we need to analyze each term to see if it can be written as the cube of some other term.
1. Expression: [tex]\(-3\)[/tex]
- The number [tex]\(-3\)[/tex] is not a perfect cube. Even though negative numbers can be perfect cubes, [tex]\(-3\)[/tex] does not fit this criterion.
2. Expression: [tex]\(125a^6\)[/tex]
- To determine if [tex]\(125a^6\)[/tex] is a perfect cube, we break it down into its components:
- [tex]\(125\)[/tex]: To see if 125 is a perfect cube, we note that [tex]\(125 = 5^3\)[/tex], so 125 is a perfect cube.
- [tex]\(a^6\)[/tex]: This can be rewritten as [tex]\((a^2)^3\)[/tex], which shows it is a perfect cube.
- Therefore, [tex]\(125a^6\)[/tex] is a perfect cube.
3. Expression: [tex]\(8m^{20}\)[/tex]
- To determine if [tex]\(8m^{20}\)[/tex] is a perfect cube, we break it down into its components:
- [tex]\(8\)[/tex]: To see if 8 is a perfect cube, we note that [tex]\(8 = 2^3\)[/tex], so 8 is a perfect cube.
- [tex]\(m^{20}\)[/tex]: We need [tex]\(m^{20}\)[/tex] to be in the form [tex]\( (some\_term)^3 \)[/tex]. However, [tex]\(20 \mod 3 \neq 0\)[/tex], thus [tex]\(m^{20}\)[/tex] cannot be expressed as a perfect cube.
- Therefore, [tex]\(8m^{20}\)[/tex] is not a perfect cube.
4. Expression: [tex]\(100w^{24}x^3\)[/tex]
- To determine if [tex]\(100w^{24}x^3\)[/tex] is a perfect cube, we break it down into its components:
- [tex]\(100\)[/tex]: To see if 100 is a perfect cube, we note that [tex]\(100 = 10^2\)[/tex], which is not a perfect cube.
- Since [tex]\(100\)[/tex] is not a perfect cube, the entire expression is therefore not a perfect cube.
5. Expression: [tex]\(-64q^{12}\)[/tex]
- To determine if [tex]\(-64q^{12}\)[/tex] is a perfect cube, we break it down into its components:
- [tex]\(-64\)[/tex]: To see if [tex]\(-64\)[/tex] is a perfect cube, we note that [tex]\(-64 = (-4)^3\)[/tex], so [tex]\(-64\)[/tex] is a perfect cube.
- [tex]\(q^{12}\)[/tex]: This can be rewritten as [tex]\((q^4)^3\)[/tex], which shows it is a perfect cube.
- Therefore, [tex]\(-64q^{12}\)[/tex] is a perfect cube.
6. Expression: [tex]\(-c^{30}d^{9}f^{18}g^{3}\)[/tex]
- To determine if [tex]\(-c^{30}d^{9}f^{18}g^{3}\)[/tex] is a perfect cube, we break it down:
- Each power in the expression:
- [tex]\(c^{30}\)[/tex] can be written as [tex]\((c^{10})^3\)[/tex]
- [tex]\(d^{9}\)[/tex] can be written as [tex]\((d^3)^3\)[/tex]
- [tex]\(f^{18}\)[/tex] can be written as [tex]\((f^6)^3\)[/tex]
- [tex]\(g^3\)[/tex] can be written as [tex]\((g)^3\)[/tex]
- Since all the components are cubes, the entire expression [tex]\(-c^{30}d^{9}f^{18}g^{3}\)[/tex] is a perfect cube.
Based on this analysis, the expressions that are perfect cubes are:
[tex]\[ 125a^6, -64q^{12}, -c^{30}d^{9}f^{18}g^3 \][/tex]
1. Expression: [tex]\(-3\)[/tex]
- The number [tex]\(-3\)[/tex] is not a perfect cube. Even though negative numbers can be perfect cubes, [tex]\(-3\)[/tex] does not fit this criterion.
2. Expression: [tex]\(125a^6\)[/tex]
- To determine if [tex]\(125a^6\)[/tex] is a perfect cube, we break it down into its components:
- [tex]\(125\)[/tex]: To see if 125 is a perfect cube, we note that [tex]\(125 = 5^3\)[/tex], so 125 is a perfect cube.
- [tex]\(a^6\)[/tex]: This can be rewritten as [tex]\((a^2)^3\)[/tex], which shows it is a perfect cube.
- Therefore, [tex]\(125a^6\)[/tex] is a perfect cube.
3. Expression: [tex]\(8m^{20}\)[/tex]
- To determine if [tex]\(8m^{20}\)[/tex] is a perfect cube, we break it down into its components:
- [tex]\(8\)[/tex]: To see if 8 is a perfect cube, we note that [tex]\(8 = 2^3\)[/tex], so 8 is a perfect cube.
- [tex]\(m^{20}\)[/tex]: We need [tex]\(m^{20}\)[/tex] to be in the form [tex]\( (some\_term)^3 \)[/tex]. However, [tex]\(20 \mod 3 \neq 0\)[/tex], thus [tex]\(m^{20}\)[/tex] cannot be expressed as a perfect cube.
- Therefore, [tex]\(8m^{20}\)[/tex] is not a perfect cube.
4. Expression: [tex]\(100w^{24}x^3\)[/tex]
- To determine if [tex]\(100w^{24}x^3\)[/tex] is a perfect cube, we break it down into its components:
- [tex]\(100\)[/tex]: To see if 100 is a perfect cube, we note that [tex]\(100 = 10^2\)[/tex], which is not a perfect cube.
- Since [tex]\(100\)[/tex] is not a perfect cube, the entire expression is therefore not a perfect cube.
5. Expression: [tex]\(-64q^{12}\)[/tex]
- To determine if [tex]\(-64q^{12}\)[/tex] is a perfect cube, we break it down into its components:
- [tex]\(-64\)[/tex]: To see if [tex]\(-64\)[/tex] is a perfect cube, we note that [tex]\(-64 = (-4)^3\)[/tex], so [tex]\(-64\)[/tex] is a perfect cube.
- [tex]\(q^{12}\)[/tex]: This can be rewritten as [tex]\((q^4)^3\)[/tex], which shows it is a perfect cube.
- Therefore, [tex]\(-64q^{12}\)[/tex] is a perfect cube.
6. Expression: [tex]\(-c^{30}d^{9}f^{18}g^{3}\)[/tex]
- To determine if [tex]\(-c^{30}d^{9}f^{18}g^{3}\)[/tex] is a perfect cube, we break it down:
- Each power in the expression:
- [tex]\(c^{30}\)[/tex] can be written as [tex]\((c^{10})^3\)[/tex]
- [tex]\(d^{9}\)[/tex] can be written as [tex]\((d^3)^3\)[/tex]
- [tex]\(f^{18}\)[/tex] can be written as [tex]\((f^6)^3\)[/tex]
- [tex]\(g^3\)[/tex] can be written as [tex]\((g)^3\)[/tex]
- Since all the components are cubes, the entire expression [tex]\(-c^{30}d^{9}f^{18}g^{3}\)[/tex] is a perfect cube.
Based on this analysis, the expressions that are perfect cubes are:
[tex]\[ 125a^6, -64q^{12}, -c^{30}d^{9}f^{18}g^3 \][/tex]
