Let's solve the problems one by one.

1. Find the value of $5^{-3} \times \frac{1}{5^3}$.

Using the properties of exponents: $5^{-3} \times \frac{1}{5^3} = 5^{-3} \times 5^{-3} = 5^{-6}$ Therefore, $5^{-6} = \frac{1}{5^6}$.

2. Simplify $\frac{2^5}{2^{-6}}$.

Using the properties of exponents, specifically $\frac{a^m}{a^n} = a^{m-n}$: $\frac{2^5}{2^{-6}} = 2^{5 - (-6)} = 2^{5+6} = 2^{11}$

3. Express $4^{-3}$ as a power with base 2.

We know that $4 = 2^2$, so: $4^{-3} = (2^2)^{-3} = 2^{2 \times (-3)} = 2^{-6}$

4. Simplify and write the answer in exponential form: $\left(\frac{2^5}{2^8}\right)^5 \times 2^5$.

First simplify inside the parentheses: $\frac{2^5}{2^8} = 2^{5-8} = 2^{-3}$ Now raise it to the power of 5: $(2^{-3})^5 = 2^{-15}$ Now multiply by $2^5$: $2^{-15} \times 2^5 = 2^{-15+5} = 2^{-10}$

5. Find $m$ such that $(-3)^{m+1} \times (-3)^5 = (-3)^7$.

Using the property $a^m \times a^n = a^{m+n}$: $(-3)^{m+1} \times (-3)^5 = (-3)^{(m+1) + 5} = (-3)^{m+6}$ Now set this equal to $(-3)^7$: $m + 6 = 7$ Solving for $m$, we get: $m = 1$

6. Find the value of $\left(\frac{2}{3}\right)^{-2}$.

Using the rule $\left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^n$: $\left(\frac{2}{3}\right)^{-2} = \left(\frac{3}{2}\right)^2 = \frac{9}{4}$

7. Simplify $\left(\frac{5}{8}\right)^{-7} \times \left(\frac{8}{5}\right)^{-5}$.

We simplify both terms: $\left(\frac{5}{8}\right)^{-7} = \left(\frac{8}{5}\right)^7$ $\left(\frac{8}{5}\right)^{-5} = \left(\frac{5}{8}\right)^5$ Now multiply the two results: $\left(\frac{8}{5}\right)^7 \times \left(\frac{5}{8}\right)^5 = \frac{(8^7)(5^5)}{(5^7)(8^5)} = \frac{8^2}{5^2} = \left(\frac{8}{5}\right)^2$

8. Simplify $(-4)^{-10} \times (-4)^5$.

Using the property $a^m \times a^n = a^{m+n}$: $(-4)^{-10} \times (-4)^5 = (-4)^{-10+5} = (-4)^{-5}$ Therefore, $(-4)^{-5} = \frac{1}{(-4)^5}$.

Do you need further clarifications on any of the steps?

Here are five related questions to deepen your understanding:

1. What is the general rule for multiplying terms with the same base and different exponents?
2. How can you express $9^{-3}$ using base 3?
3. What happens when you raise a negative base to an even power versus an odd power?
4. How do fractional exponents relate to roots, and can you give an example?
5. What is the reciprocal of $(a/b)^n$, and how is it simplified?

Tip: When dealing with exponents, remember that adding exponents occurs during multiplication, while subtracting exponents happens when dividing terms with the same base.