Let's go through each part of the exercise in the image.

Exercise 1

1. Rewrite the expressions as a power with a positive exponent:

   • A = $8^2 \times 32^2$

     • Rewrite $8$ and $32$ as powers of 2: $8 = 2^3$ and $32 = 2^5$.
     • Then $A = (2^3)^2 \times (2^5)^2 = 2^6 \times 2^{10} = 2^{16}$.

   • B = $2^{-8} \times 4^8$

     • Rewrite $4$ as $2^2$: $B = 2^{-8} \times (2^2)^8 = 2^{-8} \times 2^{16} = 2^8$.

   • C = $(5^{-3})^2 \times \frac{10^6}{64} \times 0.00001$

     • Simplify each term separately: $(5^{-3})^2 = 5^{-6}$.
     • Rewrite $64$ as $2^6$: $\frac{10^6}{64} = \frac{10^6}{2^6}$.
     • Rewrite $0.00001$ as $10^{-5}$.
     • Combine terms as needed.

   • D = $\left( \frac{2}{3} \right)^{-11} \times 3^{-2} \times 6^{11}$

     • Rewrite $6$ as $2 \times 3$, then $6^{11} = (2 \times 3)^{11} = 2^{11} \times 3^{11}$.
     • Combine powers of 2 and 3 accordingly.

   • E = $(-5)^{21} \times 125^{-7}$

     • Rewrite $125$ as $5^3$, then $125^{-7} = (5^3)^{-7} = 5^{-21}$.
     • Combine the terms accordingly.

2. Calculate the following expressions:

   • A = $\left( \frac{1}{3} \right)^{-1} - \left( \frac{5}{7} - \frac{3}{14} \right)^{-2} \cdot 99$

     • Simplify each component individually and then combine.

   • B = $5 - \left[ 1 - \left( \frac{3}{2} - 2^{-1} \right) \right]^2$

     • Simplify the inner terms and then square and subtract as required.

3. Simplify the following expressions:

   • C = $\frac{7^{100} \times 8^{24} \times 2^{41}}{125^{17} \times 49^{50} \times 5^{70}}$

     • Rewrite $8, 49, 125$ in terms of prime bases and simplify accordingly.

   • D = $32^{12} - 16^{15}$

     • Rewrite $32$ and $16$ as powers of 2 and simplify.

4. Provide the scientific notation for the following:

   • F = 2300000 × $10^{-12}$

     • Convert to scientific notation: $F = 2.3 \times 10^{-6}$.

   • G = 5.87 × $10^{-9}$ + 0.03 × $10^{-6}$

     • Convert $0.03 \times 10^{-6}$ to scientific notation and add.

Would you like further details on any specific part or have any questions?

Expanded Questions:

1. How can we simplify expressions involving negative exponents?
2. What is the method for converting large or small numbers into scientific notation?
3. Why do we rewrite numbers like 32 and 125 in terms of powers of smaller primes?
4. What are the rules for multiplying and dividing terms with exponents?
5. How can we solve expressions involving mixed operations within parentheses and powers?

Tip: Always rewrite large or composite numbers in terms of their prime factors when working with exponents; this simplifies calculations significantly.