问题1782: 初中数学四则运算

Let's evaluate each expression step by step:

(1) \((-2)^2 \times (+5)^2\)

• \((-2)^2 = 4\) (since negative squared is positive)
• \((+5)^2 = 25\)
• So, \(4 \times 25 = 100\)
  Answer: \(\boxed{100}\)

(2) \((-2)^3 \times (+5)^2\)

• \((-2)^3 = -8\) (negative cubed is negative)
• \((+5)^2 = 25\)
• So, \(-8 \times 25 = -200\)
  Answer: \(\boxed{-200}\)

(3) \((-2)^2 \times (+5)^3\)

• \((-2)^2 = 4\)
• \((+5)^3 = 125\)
• So, \(4 \times 125 = 500\)
  Answer: \(\boxed{500}\)

(4) \((-2)^6 \times (-2)\)

• \((-2)^6 = 64\) (even exponent gives positive)
• \((-2) = -2\)
• So, \(64 \times (-2) = -128\)
  Answer: \(\boxed{-128}\)

(5) \(\left( -\frac{1}{3} \right)^5 \times (-3)^5\)

• \(\left( -\frac{1}{3} \right)^5 = -\frac{1}{243}\) (odd exponent gives negative)
• \((-3)^5 = -243\) (odd exponent gives negative)
• So, \(-\frac{1}{243} \times (-243) = \frac{243}{243} = 1\)
  Answer: \(\boxed{1}\)

(6) \(\left( -\frac{1}{2} \right)^6 \times (-2)^8\)

• \(\left( -\frac{1}{2} \right)^6 = \frac{1}{64}\) (even exponent gives positive)
• \((-2)^8 = 256\) (even exponent gives positive)
• So, \(\frac{1}{64} \times 256 = \frac{256}{64} = 4\)
  Answer: \(\boxed{4}\)

(7) \(\left( -1 \frac{1}{2} \right)^2 \times \left( -\frac{2}{3} \right)^3\)

• Convert mixed number: \(-1 \frac{1}{2} = -\frac{3}{2}\)
• \(\left( -\frac{3}{2} \right)^2 = \frac{9}{4}\) (even exponent gives positive)
• \(\left( -\frac{2}{3} \right)^3 = -\frac{8}{27}\) (odd exponent gives negative)
• So, \(\frac{9}{4} \times \left(-\frac{8}{27}\right) = -\frac{72}{108} = -\frac{2}{3}\) (simplify: divide numerator and denominator by 36)
  Answer: \(\boxed{-\frac{2}{3}}\)

(8) \(\left( -1 \frac{1}{2} \right)^3 \times \left( -\frac{2}{3} \right)^4\)

• Convert mixed number: \(-1 \frac{1}{2} = -\frac{3}{2}\)
• \(\left( -\frac{3}{2} \right)^3 = -\frac{27}{8}\) (odd exponent gives negative)
• \(\left( -\frac{2}{3} \right)^4 = \frac{16}{81}\) (even exponent gives positive)
• So, \(-\frac{27}{8} \times \frac{16}{81} = -\frac{432}{648} = -\frac{2}{3}\) (simplify: divide numerator and denominator by 216)
  Answer: \(\boxed{-\frac{2}{3}}\)

Summary of Answers:

1. \(\boxed{100}\)
2. \(\boxed{-200}\)
3. \(\boxed{500}\)
4. \(\boxed{-128}\)
5. \(\boxed{1}\)
6. \(\boxed{4}\)
7. \(\boxed{-\frac{2}{3}}\)
8. \(\boxed{-\frac{2}{3}}\)

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