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It looks like you’ve uploaded an image of a math worksheet focusing on exponent rules. I will extract the questions from the image and provide solutions for you. Let me begin by analyzing and solving each problem.

### Problems identified:
1. \( \left( \frac{1}{8} \right)^5 \times \left( \frac{1}{8} \right)^3 \)
2. \( 3k^5 \times 4k^5 \)
3. \( \frac{45k^7}{5k^3} \)
4. \( \frac{18h^8}{6h^3} \)
5. \( (3d^7)^2 \)
6. \( (2z^7 \cdot 5z)^2 \)
7. \( \frac{3}{r^4} \)
8. \( 2c^0 = 9 \)
9. \( \frac{8n^7}{x^3} \)
10. \( (x^7y^2z)^2 \)

I'll work through the solutions for each one.

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1. \( \left( \frac{1}{8} \right)^5 \times \left( \frac{1}{8} \right)^3 \)

Use the rule of exponents for multiplication: \( a^m \times a^n = a^{m+n} \).

\[
\left( \frac{1}{8} \right)^5 \times \left( \frac{1}{8} \right)^3 = \left( \frac{1}{8} \right)^{5+3} = \left( \frac{1}{8} \right)^8 = \frac{1}{8^8}
\]

2. \( 3k^5 \times 4k^5 \)

Multiply the constants first, and then apply the exponent rule:

\[
3 \times 4 = 12 \quad \text{and} \quad k^5 \times k^5 = k^{5+5} = k^{10}
\]

So the answer is:

\[
12k^{10}
\]

3. \( \frac{45k^7}{5k^3} \)

First, divide the constants:

\[
\frac{45}{5} = 9
\]

Then apply the exponent rule for division \( a^m / a^n = a^{m-n} \):

\[
k^{7-3} = k^4
\]

So, the result is:

\[
9k^4
\]

4. \( \frac{18h^8}{6h^3} \)

Divide the constants:

\[
\frac{18}{6} = 3
\]

Then apply the exponent rule:

\[
h^{8-3} = h^5
\]

So the answer is:

\[
3h^5
\]

5. \( (3d^7)^2 \)

Apply the power rule \( (a^m)^n = a^{m \cdot n} \):

\[
(3d^7)^2 = 3^2 \cdot d^{7 \times 2} = 9d^{14}
\]

6. \( (2z^7 \cdot 5z)^2 \)

First simplify inside the parentheses:

\[
2z^7 \cdot 5z = (2 \times 5)z^{7+1} = 10z^8
\]

Now apply the power of a product rule:

\[
(10z^8)^2 = 10^2 \cdot z^{8 \times 2} = 100z^{16}
\]

7. \( \frac{3}{r^4} \)

This is already simplified as a fraction, and there are no further steps required.

8. \( 2c^0 = 9 \)

Recall that any number raised to the power of 0 is 1:

\[
c^0 = 1
\]

So, this equation simplifies to:

\[
2 \times 1 = 2
\]

This contradicts the equation \( 2 = 9 \), so there is **no solution** to this equation.

9. \( \frac{8n^7}{x^3} \)

This expression is already simplified. So the result is:

\[
\frac{8n^7}{x^3}
\]

10. \( (x^7y^2z)^2 \)

Apply the power rule to each factor:

\[
(x^7y^2z)^2 = x^{7 \times 2} \cdot y^{2 \times 2} \cdot z^{1 \times 2} = x^{14}y^4z^2
\]

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### Let me know if you want more details on any of the solutions!

#### 5 Related Questions:
1. How do you handle negative exponents in multiplication or division?
2. What happens when multiplying or dividing powers with the same base but different exponents?
3. Can you explain the process for simplifying fractions with variables in both the numerator and the denominator?
4. How do you solve equations involving powers of zero?
5. What are the rules for applying exponents to products and quotients?

**Tip:** Always simplify expressions inside parentheses before applying exponent rules to avoid mistakes!

Math worksheet covering exponent rules with various problems involving powers, multiplication, and division of expressions with exponents.

This worksheet covers exponent rules for simplifying expressions involving powers, multiplication, division, and zero exponents. Step-by-step solutions to problems like (3k^5) * (4k^5) and (x^7y^2z)^2 are provided. Suitable for Grades 7-9.

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