I will work through each of these problems with detailed explanations and formulas for simplifying them. Let's go through each part step-by-step.

Problem (a)

$2^{-2} \cdot 4^3$

Solution:

1. Rewrite $2^{-2}$ using the negative exponent rule: $a^{-n} = \frac{1}{a^n}$. $2^{-2} = \frac{1}{2^2} = \frac{1}{4}$

2. Rewrite $4^3$ as $(2^2)^3$ and apply the power of a power rule: $(a^m)^n = a^{m \cdot n}$. $4^3 = (2^2)^3 = 2^{2 \cdot 3} = 2^6$

3. Combine the terms: $2^{-2} \cdot 4^3 = \frac{1}{4} \cdot 2^6$

4. Rewrite $\frac{1}{4}$ as $2^{-2}$, then combine exponents. $2^{-2} \cdot 2^6 = 2^{(-2 + 6)} = 2^4 = 16$

Answer for (a): $16$

Problem (b)

$\frac{3^{19}}{9^5}$

Solution:

1. Rewrite $9^5$ as $(3^2)^5$ and apply the power of a power rule. $9^5 = (3^2)^5 = 3^{2 \cdot 5} = 3^{10}$

2. Substitute into the expression: $\frac{3^{19}}{3^{10}}$

3. Use the division rule for exponents: $\frac{a^m}{a^n} = a^{m-n}$. $3^{19 - 10} = 3^9$

Answer for (b): $3^9$

Problem (c)

$10^3 \cdot 5^4$

Solution:

This expression is already simplified since $10^3$ and $5^4$ are in their simplest forms and do not have common bases. The answer remains as is.

Answer for (c): $10^3 \cdot 5^4$

Problem (d)

$5^7 \cdot \frac{2}{10^2}$

Solution:

1. Rewrite $10^2$ as $(2 \cdot 5)^2 = 2^2 \cdot 5^2$. $10^2 = 2^2 \cdot 5^2$

2. Substitute into the expression: $5^7 \cdot \frac{2}{2^2 \cdot 5^2}$

3. Simplify by canceling terms:

   • For $2$ in the numerator and $2^2$ in the denominator: $\frac{2}{2^2} = 2^{1-2} = 2^{-1} = \frac{1}{2}$.
   • For $5^7$ and $5^2$: $\frac{5^7}{5^2} = 5^{7-2} = 5^5$.

4. Combine the simplified terms: $5^5 \cdot \frac{1}{2} = \frac{5^5}{2}$

Answer for (d): $\frac{5^5}{2}$

Would you like further explanations on any of these solutions?

Here are five related questions to expand your understanding:

1. How would you simplify an expression involving a fractional exponent, such as $x^{\frac{3}{2}}$?
2. What is the result of $(3^{-2})^4$?
3. How can you simplify expressions involving roots and exponents, like $\sqrt[3]{x^6}$?
4. Can you express $8^2 \cdot 4^{-3}$ using a single base?
5. What are the steps to simplify expressions with mixed exponents and roots, such as $9^{\frac{3}{2}}$?

Tip: Remember that converting all terms to the same base when possible makes simplification easier and helps avoid mistakes with exponents.