Let's solve each part of the problem step-by-step.

Given Expressions:

1. (i) $(-2)^7$
2. (ii) $(-2)^8$
3. (iii) $(-4)^8$
4. (iv) $(-3)^2 \times (-5)^3$
5. (v) $(-2)^3 \times (-5)^3$
6. (vi) $(-3) \times (-2)^5$

Solutions:

(i) $(-2)^7$

$(-2)^7 = -128 \quad (\text{as the power is odd, the result is negative})$

(ii) $(-2)^8$

$(-2)^8 = 256 \quad (\text{as the power is even, the result is positive})$

(iii) $(-4)^8$

$(-4)^8 = 4^8 = 65536 \quad (\text{as the power is even, the result is positive})$

(iv) $(-3)^2 \times (-5)^3$

1. First, calculate each term:
   • $(-3)^2 = 9$ (even power, positive result)
   • $(-5)^3 = -125$ (odd power, negative result)
2. Multiply: $9 \times (-125) = -1125$

(v) $(-2)^3 \times (-5)^3$

1. Calculate each term:
   • $(-2)^3 = -8$ (odd power, negative result)
   • $(-5)^3 = -125$ (odd power, negative result)
2. Multiply: $(-8) \times (-125) = 1000$

(vi) $(-3) \times (-2)^5$

1. Calculate each term:
   • $(-3) = -3$
   • $(-2)^5 = -32$ (odd power, negative result)
2. Multiply: $(-3) \times (-32) = 96$

Final Answers:

1. (i) $(-2)^7 = -128$
2. (ii) $(-2)^8 = 256$
3. (iii) $(-4)^8 = 65536$
4. (iv) $(-3)^2 \times (-5)^3 = -1125$
5. (v) $(-2)^3 \times (-5)^3 = 1000$
6. (vi) $(-3) \times (-2)^5 = 96$

Would you like a detailed explanation for any part? Here are 5 related questions:

1. How do powers of negative numbers change based on even or odd exponents?
2. Can you explain why $(-2)^3$ and $(-2)^8$ produce different signs?
3. How do you calculate the product of two negative terms, like in (v)?
4. What happens if the base is positive but the exponent is negative?
5. Can you apply similar rules to simplify fractions with negative bases?

Tip: Always determine the sign first when working with powers of negative numbers to avoid