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Sagot :
Sure, let's simplify each expression step-by-step.
### Simplifying [tex]\((a^3 b^2)^4\)[/tex]
1. Expand the powers in the expression:
[tex]\[ (a^3 b^2)^4 \][/tex]
2. Apply the power rule for exponents [tex]\((x^m)^n = x^{m \cdot n}\)[/tex]:
[tex]\[ (a^3)^4 \cdot (b^2)^4 \][/tex]
3. Calculate the exponents separately:
[tex]\[ a^{3 \cdot 4} \cdot b^{2 \cdot 4} \][/tex]
4. Multiply the exponents:
[tex]\[ a^{12} \cdot b^{8} \][/tex]
Thus, the simplified form of [tex]\((a^3 b^2)^4\)[/tex] is:
[tex]\[ a^{12} b^8 \][/tex]
### Simplifying [tex]\((2 x^2 y)^3 \cdot (4 + y^3)^2\)[/tex]
1. Expand the first part [tex]\((2 x^2 y)^3\)[/tex]:
[tex]\[ (2 x^2 y)^3 \][/tex]
2. Apply the power rule for exponents:
[tex]\[ 2^3 \cdot (x^2)^3 \cdot y^3 \][/tex]
3. Calculate the exponents separately and the constant power:
[tex]\[ 8 \cdot x^{2 \cdot 3} \cdot y^3 \][/tex]
4. Multiply the exponents:
[tex]\[ 8 \cdot x^6 \cdot y^3 \][/tex]
5. Now expand the second part [tex]\((4 + y^3)^2\)[/tex]:
[tex]\[ (4 + y^3)^2 \][/tex]
6. We treat this as a binomial expression and square it directly as it stands.
Thus, we have:
[tex]\[ (8 x^6 y^3) \cdot (4 + y^3)^2 \][/tex]
So, the final simplified form of [tex]\((2 x^2 y)^3 \cdot (4 + y^3)^2\)[/tex] is:
[tex]\[ 8 x^6 y^3 (4 + y^3)^2 \][/tex]
Thus, the complete simplified expressions are:
[tex]\[ \begin{array}{l} (a^3 b^2)^4 = a^{12} b^8, \\ (2 x^2 y)^3 \cdot (4 + y^3)^2 = 8 x^6 y^3 (4 + y^3)^2 \end{array} \][/tex]
### Simplifying [tex]\((a^3 b^2)^4\)[/tex]
1. Expand the powers in the expression:
[tex]\[ (a^3 b^2)^4 \][/tex]
2. Apply the power rule for exponents [tex]\((x^m)^n = x^{m \cdot n}\)[/tex]:
[tex]\[ (a^3)^4 \cdot (b^2)^4 \][/tex]
3. Calculate the exponents separately:
[tex]\[ a^{3 \cdot 4} \cdot b^{2 \cdot 4} \][/tex]
4. Multiply the exponents:
[tex]\[ a^{12} \cdot b^{8} \][/tex]
Thus, the simplified form of [tex]\((a^3 b^2)^4\)[/tex] is:
[tex]\[ a^{12} b^8 \][/tex]
### Simplifying [tex]\((2 x^2 y)^3 \cdot (4 + y^3)^2\)[/tex]
1. Expand the first part [tex]\((2 x^2 y)^3\)[/tex]:
[tex]\[ (2 x^2 y)^3 \][/tex]
2. Apply the power rule for exponents:
[tex]\[ 2^3 \cdot (x^2)^3 \cdot y^3 \][/tex]
3. Calculate the exponents separately and the constant power:
[tex]\[ 8 \cdot x^{2 \cdot 3} \cdot y^3 \][/tex]
4. Multiply the exponents:
[tex]\[ 8 \cdot x^6 \cdot y^3 \][/tex]
5. Now expand the second part [tex]\((4 + y^3)^2\)[/tex]:
[tex]\[ (4 + y^3)^2 \][/tex]
6. We treat this as a binomial expression and square it directly as it stands.
Thus, we have:
[tex]\[ (8 x^6 y^3) \cdot (4 + y^3)^2 \][/tex]
So, the final simplified form of [tex]\((2 x^2 y)^3 \cdot (4 + y^3)^2\)[/tex] is:
[tex]\[ 8 x^6 y^3 (4 + y^3)^2 \][/tex]
Thus, the complete simplified expressions are:
[tex]\[ \begin{array}{l} (a^3 b^2)^4 = a^{12} b^8, \\ (2 x^2 y)^3 \cdot (4 + y^3)^2 = 8 x^6 y^3 (4 + y^3)^2 \end{array} \][/tex]
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