1. **Problem:** Calculate $\log(1000 \times 1000)$. **Step 1:** Use the logarithm product rule: $\log(ab) = \log a + \log b$. **Step 2:** Calculate $\log(1000)$. Since $1000 = 10^3$, $\log(1000) = 3$. **Step 3:** Therefore, $\log(1000 \times 1000) = \log(1000) + \log(1000) = 3 + 3 = 6$. 2. **Problem:** Calculate $\ln e^2$. **Step 1:** Use the logarithm power rule: $\ln e^x = x$. **Step 2:** So, $\ln e^2 = 2$. 3. **Problem:** Find the set $A = \{|x|; 20 < x < 50; x = \text{prime number}\}$. **Step 1:** Identify prime numbers between 20 and 50: 23, 29, 31, 37, 41, 43, 47. **Step 2:** Take absolute values (all positive here), so $A = \{23, 29, 31, 37, 41, 43, 47\}$. 4. **Problem:** Solve the inequality $2x + 2 \leq 12$. **Step 1:** Subtract 2 from both sides: $2x \leq 10$. **Step 2:** Divide both sides by 2: $x \leq 5$. **Step 3:** Interval notation: $(-\infty, 5]$. 5. **Problem:** Solve the inequality $2 < x + 9 < 15$. **Step 1:** Subtract 9 from all parts: $2 - 9 < x < 15 - 9$. **Step 2:** Simplify: $-7 < x < 6$. **Step 3:** Interval notation: $(-7, 6)$. 6. **Problem:** Calculate $(-5 - 7i) - (-9 + 2i) + (-4 + 2i) - (-2 + 5i)$. **Step 1:** Remove parentheses carefully: $= -5 - 7i + 9 - 2i - 4 + 2i + 2 - 5i$. **Step 2:** Combine real parts: $(-5 + 9 - 4 + 2) = 2$. **Step 3:** Combine imaginary parts: $(-7i - 2i + 2i - 5i) = -12i$. **Step 4:** Final result: $2 - 12i$. 7. **Problem:** Calculate $\frac{(1 + 3i)(2 - 2i)}{(2 + i)(1 - i)}$. **Step 1:** Multiply numerator: $(1)(2) + (1)(-2i) + (3i)(2) + (3i)(-2i) = 2 - 2i + 6i - 6i^2$. **Step 2:** Since $i^2 = -1$, $-6i^2 = 6$. **Step 3:** Numerator simplifies to $2 - 2i + 6i + 6 = 8 + 4i$. **Step 4:** Multiply denominator: $(2)(1) + (2)(-i) + (i)(1) + (i)(-i) = 2 - 2i + i - i^2$. **Step 5:** Since $i^2 = -1$, $-i^2 = 1$. **Step 6:** Denominator simplifies to $2 - 2i + i + 1 = 3 - i$. **Step 7:** Divide complex numbers: $$\frac{8 + 4i}{3 - i} = \frac{(8 + 4i)(3 + i)}{(3 - i)(3 + i)}$$ **Step 8:** Multiply numerator: $8 \times 3 + 8 \times i + 4i \times 3 + 4i \times i = 24 + 8i + 12i + 4i^2 = 24 + 20i + 4(-1) = 24 + 20i - 4 = 20 + 20i$. **Step 9:** Multiply denominator: $3^2 - (i)^2 = 9 - (-1) = 10$. **Step 10:** Final division: $$\frac{20 + 20i}{10} = 2 + 2i$$. 8. **Problem:** Calculate $(2 - 2i)^5$ using the binomial theorem. **Step 1:** Express in polar form: Magnitude $r = \sqrt{2^2 + (-2)^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2}$. **Step 2:** Argument $\theta = \arctan\left(\frac{-2}{2}\right) = \arctan(-1) = -\frac{\pi}{4}$. **Step 3:** Use De Moivre's theorem: $$(2 - 2i)^5 = r^5 \left(\cos(5\theta) + i \sin(5\theta)\right)$$ **Step 4:** Calculate $r^5 = (2\sqrt{2})^5 = 2^5 \times (\sqrt{2})^5 = 32 \times 2^{5/2} = 32 \times 2^{2.5} = 32 \times 5.656854 = 181.0193$ (approx). **Step 5:** Calculate $5\theta = 5 \times -\frac{\pi}{4} = -\frac{5\pi}{4}$. **Step 6:** Calculate $\cos(-\frac{5\pi}{4}) = \cos(\frac{5\pi}{4}) = -\frac{\sqrt{2}}{2}$ and $\sin(-\frac{5\pi}{4}) = -\sin(\frac{5\pi}{4}) = \frac{\sqrt{2}}{2}$. **Step 7:** Final result: $$181.0193 \times \left(-\frac{\sqrt{2}}{2} + i \frac{\sqrt{2}}{2}\right) = -128 + 128i$$ (rounded to nearest integer).