Let's go through the problems from the image and solve them one by one:

4. Find the value of:

$\frac{0.8a^5(b+c)^2c^2(b+c)^{-2}}{1 \times 10^9}$

• First, simplify the expression inside the numerator: $a^5(b+c)^2c^2(b+c)^{-2} = a^5c^2(b+c)^{2-2} = a^5c^2(b+c)^0 = a^5c^2$
• Now, the expression becomes: $\frac{0.8a^5c^2}{1 \times 10^9}$
• Convert $0.8$ to scientific notation: $0.8 = 8 \times 10^{-1}$ The expression now is: $\frac{8 \times 10^{-1} a^5 c^2}{1 \times 10^9} = 8a^5c^2 \times 10^{-1-9} = 8a^5c^2 \times 10^{-10}$
• Hence, the correct answer is c. 8a^5c^2.

5. Simplify:

$\frac{25^{-4} \times 625^{-3}}{(125^{-1})^{-3}} = \ldots$

• Start by simplifying the terms: $625 = 25^2 \quad \text{so} \quad 625^{-3} = (25^2)^{-3} = 25^{-6}$ $125 = 5^3 \quad \text{so} \quad (125^{-1})^{-3} = (5^3)^{-1 \times -3} = 5^9$
• Now, the expression becomes: $\frac{25^{-4} \times 25^{-6}}{5^9} = \frac{25^{-10}}{5^9}$
• Since $25 = 5^2$, we can write: $25^{-10} = (5^2)^{-10} = 5^{-20}$ The expression becomes: $\frac{5^{-20}}{5^9} = 5^{-20-9} = 5^{-29}$
• Hence, the correct answer is d. 5^{-29}.

6. Simplify:

$4^{-2} \times 4^4 \times 4^{-3} = \ldots$

• Using the property of exponents: $4^{-2} \times 4^4 \times 4^{-3} = 4^{-2+4-3} = 4^{-1}$
• Hence, the correct answer is d. 4^{-1}.

7. Simplify the following expression:

$\frac{a^6b^6c^6}{c^{-4}b^{-3}a^2} = \ldots$

• Simplify each variable's exponents: $a^6 \div a^2 = a^{6-2} = a^4$ $b^6 \div b^{-3} = b^{6+3} = b^9$ $c^6 \div c^{-4} = c^{6+4} = c^{10}$
• Hence, the simplified expression is: $a^4b^9c^{10}$
• Therefore, the correct answer is a. a^4b^9c^{10}.

Would you like further clarification on any of the steps?

Here are 5 related questions to deepen your understanding:

1. How would you simplify an expression like $(x^2y^{-3})(x^{-1}y^4)$?
2. What is the result of dividing powers of the same base, such as $\frac{a^7}{a^3}$?
3. How do you handle negative exponents when simplifying expressions?
4. Can you explain how scientific notation affects the calculation in problem 4?
5. What is the general rule for multiplying powers of the same base?

Tip: When dealing with powers and roots, always try to express all terms with the same base to make simplification easier!