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решай только номера 692,694,695 под нечетными номерами

Let's solve the odd-numbered exercises from the specified problems: 692, 694, and 695.

### Problem 692 (1st exercise):
\[ 7x^7 - 7x = 6 \]

**Solution:**
1. Factor out \( 7x \):
   \[
   7x(x^6 - 1) = 6
   \]
2. Divide both sides by 7:
   \[
   x(x^6 - 1) = \frac{6}{7}
   \]
3. Expand \( x^6 - 1 \) and simplify:
   \[
   x^7 - x = \frac{6}{7}
   \]
4. This is a non-trivial equation involving powers of \( x \). It cannot be solved algebraically here without specific approximations or numerical methods.

---

### Problem 694 (1st exercise):
\[ 8 \cdot 4^{1 - x} - 12 \cdot 2^x = 0 \]

**Solution:**
1. Rewrite \( 4^{1 - x} \) as \( (2^2)^{1 - x} = 2^{2(1 - x)} \):
   \[
   8 \cdot 2^{2(1 - x)} - 12 \cdot 2^x = 0
   \]
2. Simplify:
   \[
   8 \cdot 2^{2 - 2x} = 12 \cdot 2^x
   \]
3. Divide both sides by 4:
   \[
   2 \cdot 2^{2 - 2x} = 3 \cdot 2^x
   \]
4. Combine powers of 2 on both sides:
   \[
   2^{3 - 2x} = 3 \cdot 2^x
   \]
5. Since this involves powers of 2 and 3, solve using logarithms or approximate numerically for \( x \).

---

### Problem 695 (1st exercise):
\[ 2^x + 2^{x + 2} = 22 \]

**Solution:**
1. Factor out \( 2^x \):
   \[
   2^x(1 + 2^2) = 22
   \]
2. Simplify:
   \[
   2^x(1 + 4) = 22
   \]
   \[
   2^x \cdot 5 = 22
   \]
3. Divide both sides by 5:
   \[
   2^x = \frac{22}{5} = 4.4
   \]
4. Take the logarithm of both sides:
   \[
   x \log 2 = \log 4.4
   \]
5. Solve for \( x \):
   \[
   x = \frac{\log 4.4}{\log 2} \approx 2.15
   \]

Let me know if you need more details on any of these steps or a numerical solution for the remaining equations.

---

Here are some follow-up questions for you to practice:

1. What is the general method for solving exponential equations?
2. How do logarithmic properties simplify equations with powers of 2?
3. Can you solve \( 2^{2x} = 8 \cdot 2^x \) for \( x \)?
4. How can you apply factoring to solve equations like \( 3x^3 - 27x = 0 \)?
5. What is the relationship between exponential growth and logarithms in solving equations?

**Tip:** Always try to simplify exponential equations by factoring or rewriting them with common bases before applying logarithms.

Explore detailed solutions for exponential equations from problems 692, 694, and 695, focusing on odd-numbered exercises. This guide covers algebraic techniques like factoring, simplifying exponents, and using logarithms to find solutions. Ideal for students in Grades 10-12.

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