576. Упростите выражение:

Решение:

Упростим выражения, применяя свойства степеней.

1. a) \( (-x^2y^2) \cdot (-xy)^2 = (-x^2y^2) \cdot ((-1)^2x^2y^2) = (-x^2y^2) \cdot (x^2y^2) = -(x^2)^2(y^2)^2 = -x^4y^4 \)
2. б) \( -\frac{1}{3}xy^3 \cdot (-3x)^3 = -\frac{1}{3}xy^3 \cdot ((-3)^3x^3) = -\frac{1}{3}xy^3 \cdot (-27x^3) = (\frac{1}{3} \cdot 27) \cdot (x \cdot x^3) \cdot y^3 = 9x^4y^3 \)
3. в) \( (-2x^3y^2)^3 \cdot (-2y^2)^3 = ((-2)^3(x^3)^3(y^2)^3) \cdot ((-2)^3(y^2)^3) = (-8x^9y^6) \cdot (-8y^6) = (-8) \cdot (-8) \cdot x^9 \cdot (y^6 \cdot y^6) = 64x^9y^{12} \)
4. г) \( (\frac{1}{3}a^2b)^3 \cdot (9ab^2)^2 = ((\frac{1}{3})^3(a^2)^3b^3) \cdot (9^2a^2(b^2)^2) = (\frac{1}{27}a^6b^3) \cdot (81a^2b^4) = (\frac{81}{27}) \cdot (a^6a^2) \cdot (b^3b^4) = 3a^8b^7 \)
5. д) \( (-5a^3b) \cdot (\frac{1}{5}ab^3) = (-5 \cdot \frac{1}{5}) \cdot (a^3a) \cdot (b \cdot b^3) = -1a^4b^4 = -a^4b^4 \)
6. е) \( (-\frac{2}{7}ab^4)^2 \cdot (-3\frac{1}{2}a^3b)^2 = ((\frac{-2}{7})^2a^2(b^4)^2) \cdot ((\frac{-7}{2})^2(a^3)^2b^2) = (\frac{4}{49}a^2b^8) \cdot (\frac{49}{4}a^6b^2) = (\frac{4}{49} \cdot \frac{49}{4}) \cdot (a^2a^6) \cdot (b^8b^2) = 1a^8b^{10} = a^8b^{10} \)
7. ж) \( (x^3y)^2 \cdot (-5xy)^3 = (x^6y^2) \cdot ((-5)^3x^3y^3) = (x^6y^2) \cdot (-125x^3y^3) = -125 \cdot (x^6x^3) \cdot (y^2y^3) = -125x^9y^5 \)
8. з) \( (\frac{1}{6}x^2y^2) \cdot (-12x^3y^5)^2 = (\frac{1}{6}x^2y^2) \cdot ((-12)^2(x^3)^2(y^5)^2) = (\frac{1}{6}x^2y^2) \cdot (144x^6y^{10}) = (\frac{144}{6}) \cdot (x^2x^6) \cdot (y^2y^{10}) = 24x^8y^{12} \)

Ответ: a) \(-x^4y^4\); б) \(9x^4y^3\); в) \(64x^9y^{12}\); г) \(3a^8b^7\); д) \(-a^4b^4\); е) \(a^8b^{10}\); ж) \(-125x^9y^5\); з) \(24x^8y^{12}\).