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### ΘΕΜΑ 1:

You are asked to find the value of the following expression:

\[
A = \frac{-2 - 5 \cdot (-3) + (-2)^3}{-1 - 4 \cdot 2 - (-1)}
\]

Let's break it down and solve it:

#### Numerator:
\[
-2 - 5 \cdot (-3) + (-2)^3 = -2 + 15 + (-8) = 5
\]

#### Denominator:
\[
-1 - 4 \cdot 2 - (-1) = -1 - 8 + 1 = -8
\]

Thus, the value of \( A \) becomes:

\[
A = \frac{5}{-8} = -\frac{5}{8}
\]

### ΘΕΜΑ 2:

We are asked to simplify the following expressions using the properties of powers.

#### (a)
\[
\frac{(-2)^{-3} \cdot 2^2 \cdot 2^{-5}}{(-2)^6 \cdot 2^{-2}}
\]

Using the rule \( a^m \cdot a^n = a^{m+n} \), we simplify both the numerator and denominator.

##### Numerator:
\[
(-2)^{-3} \cdot 2^2 \cdot 2^{-5} = (-2)^{-3} \cdot 2^{2-5} = (-2)^{-3} \cdot 2^{-3}
\]

##### Denominator:
\[
(-2)^6 \cdot 2^{-2}
\]

The full expression becomes:

\[
\frac{(-2)^{-3} \cdot 2^{-3}}{(-2)^6 \cdot 2^{-2}} = (-2)^{-3 - 6} \cdot 2^{-3+2} = (-2)^{-9} \cdot 2^{-1}
\]

Now, we can simplify:

\[
= \frac{1}{(-2)^9 \cdot 2} = \frac{1}{-512 \cdot 2} = \frac{1}{-1024}
\]

#### (b)
\[
\frac{(-3)^3 \cdot (-4)^2}{3^3 \cdot 2^2}
\]

Simplifying both numerator and denominator:

##### Numerator:
\[
(-3)^3 \cdot (-4)^2 = -27 \cdot 16 = -432
\]

##### Denominator:
\[
3^3 \cdot 2^2 = 27 \cdot 4 = 108
\]

Thus, the expression becomes:

\[
\frac{-432}{108} = -4
\]

### ΘΕΜΑ 3:

You are asked to simplify these expressions using properties of exponents.

#### (a)
\[
\frac{(-2)^{-6} \cdot (2^2)^{-4}}{2^{-10} \cdot (-2)^7}
\]

Simplifying using the power rules:

\[
(-2)^{-6} = \frac{1}{(-2)^6}, \quad (2^2)^{-4} = 2^{-8}, \quad (-2)^7 = -128
\]

After simplification:

\[
= \frac{\frac{1}{64} \cdot 2^{-8}}{2^{-10} \cdot (-128)} = \frac{1}{64 \cdot 2^8 \cdot (-128)} = \frac{1}{-32768}
\]

#### (b)
\[
[(-1)^{2021}]^{2022}
\]

This simplifies as:

\[
(-1)^{2021 \cdot 2022} = (-1)^{4088442}
\]

Since the exponent is odd, the result is:

\[
(-1)^{\text{odd}} = -1
\]

---

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### Related Questions:
1. How can powers with negative bases be simplified when combined with exponents?
2. What are the key exponent rules needed to simplify expressions like those in ΘΕΜΑ 2?
3. How do you handle exponents that are large but have simple bases like 1 or -1?
4. What steps can be followed to break down complex fractions involving powers?
5. Why does the sign change for odd versus even powers of negative numbers?

#### Tip:
Always simplify exponents by applying the basic rules such as \( a^m \cdot a^n = a^{m+n} \), and handle negative bases carefully based on whether the exponent is odd or even!

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