Let's go through the questions in this image one by one:

1. Prove the exponent properties for numbers 6 and 7.

You need to refer to the exponent properties, such as:

• For 6, likely $a^m \times a^n = a^{m+n}$
• For 7, likely $(a^m)^n = a^{m \times n}$

For the proof, you should use the definition of exponents and basic algebraic manipulation.

2. Determine the value of $p$ so that the following equations hold true:

a. $(3^4)^2 = 3^p$

• Using the exponent rule $(a^m)^n = a^{m \times n}$, the left-hand side becomes: $(3^4)^2 = 3^{4 \times 2} = 3^8$ Therefore, $p = 8$.

b. $b^p \times b^5 = b^9$

• Using the rule $a^m \times a^n = a^{m+n}$, the left-hand side becomes: $b^p \times b^5 = b^{p+5}$ For $b^{p+5} = b^9$, we solve: $p + 5 = 9 \quad \Rightarrow \quad p = 4$

c. $(3\pi)^p = 27\pi^3$

• Rewrite $27\pi^3$ as $(3^3)\pi^3$, so the equation becomes: $(3\pi)^p = 3^3 \times \pi^3$ This suggests that $p = 3$ because the exponents of $3$ and $\pi$ match.

3. Simplify the following expressions:

a. $\left( \frac{2^4 \times 3^6}{2^3 \times 3^2} \right)^3$

• First, simplify inside the parentheses: $\frac{2^4 \times 3^6}{2^3 \times 3^2} = 2^{4-3} \times 3^{6-2} = 2^1 \times 3^4$ Now, apply the exponent rule $(a^m)^n = a^{m \times n}$: $(2^1 \times 3^4)^3 = 2^{1 \times 3} \times 3^{4 \times 3} = 2^3 \times 3^{12}$

b. $(3u^3v^5)(9u^4v)$

• Simplify by multiplying the constants and adding the exponents for the same bases: $3 \times 9 = 27$ For $u$, $u^3 \times u^4 = u^{3+4} = u^7$, and for $v$, $v^5 \times v^1 = v^{5+1} = v^6$. Thus, the result is: $27u^7v^6$

c. $\left( \frac{n^{-1} r^4}{5n^{-6} r^4} \right)^2$ where $n \neq 0, r \neq 0$

• Simplify inside the parentheses: $\frac{n^{-1} r^4}{5n^{-6} r^4} = \frac{n^{-1}}{5n^{-6}} = \frac{n^{-1}}{n^{-6} \times 5} = \frac{n^{-1} \times n^6}{5} = \frac{n^{5}}{5}$ Now, apply the exponent outside: $\left( \frac{n^{5}}{5} \right)^2 = \frac{n^{5 \times 2}}{5^2} = \frac{n^{10}}{25}$

Would you like further details or have any questions on these problems? Below are five related questions to expand on this:

1. Can you prove the exponent rule $a^m \times a^n = a^{m+n}$ with a general example?
2. How do you solve equations involving logarithms if exponents are not simple integers?
3. How can you simplify more complex expressions involving multiple bases and exponents?
4. What are some applications of exponent rules in real-world problems?
5. How do you handle equations with negative or fractional exponents?

Tip: Always check the base in exponent rules to make sure they are the same before applying properties.