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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.

1. Find the value of 5^-3 * 1/5^3. 2. Simplify 2^5 / 2^-6. 3. Express 4^-3 as a power with base 2. 4. Simplify and write the answer in exponential form: (2^5 / 2^8)^5 * 2^5. 5. Find m such that (-3)^(m+1) * (-3)^5 = (-3)^7. 6. Find the value of (2/3)^-2. 7. Simplify: (5/8)^-7 * (8/5)^-5. 8. Simplify (-4)^-10 * (-4)^5.

Learn how to solve exponent problems such as simplifying (5^-3) * (1/5^3), simplifying expressions like 2^5 / 2^-6, and more. This guide covers exponent rules and power expressions, suitable for students in Grades 9-12.

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