Let me analyze the questions from the image and work through them.

Problem 4:

We need to evaluate the expression:

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

Simplifying the numerator:

$0.8a^5(b+c)^2c^2(b+c)^{-2}$

The terms $(b+c)^2$ and $(b+c)^{-2}$ cancel each other out, leaving:

$0.8a^5c^2$

So the expression becomes:

$\frac{0.8a^5c^2}{1 \times 10^9}$

This simplifies to:

$0.8a^5c^2 \times 10^{-9}$

Multiplying $0.8$ by $10^{-9}$:

$8 \times 10^{-10} \cdot a^5c^2 = 80a^5c^2 \times 10^{-11}$

Thus, the answer is $80a^5c^2$, which corresponds to option B.

Problem 5:

We need to simplify the expression:

$\frac{25^{-4} \times 625^{-3}}{(125^{-1} \times 5^{-3})}$

First, express 25, 625, and 125 in terms of base 5:

• $25 = 5^2$, so $25^{-4} = (5^2)^{-4} = 5^{-8}$
• $625 = 5^4$, so $625^{-3} = (5^4)^{-3} = 5^{-12}$
• $125 = 5^3$, so $125^{-1} = (5^3)^{-1} = 5^{-3}$

Substitute these back into the expression:

$\frac{5^{-8} \times 5^{-12}}{5^{-3} \times 5^{-3}} = \frac{5^{-20}}{5^{-6}} = 5^{-14}$

Thus, the answer is $5^{-14}$, which corresponds to option C.

Problem 6:

We need to simplify the expression:

$\frac{4^{-2} \times 4^{-4} \times 3^{-3}}{2^{-5} \times 5^{-3}}$

First, express 4 as a power of 2:

• $4 = 2^2$, so $4^{-2} = (2^2)^{-2} = 2^{-4}$ and $4^{-4} = (2^2)^{-4} = 2^{-8}$

Substitute these into the expression:

$\frac{2^{-4} \times 2^{-8} \times 3^{-3}}{2^{-5} \times 5^{-3}} = \frac{2^{-12} \times 3^{-3}}{2^{-5} \times 5^{-3}}$

Now simplify:

$\frac{2^{-12}}{2^{-5}} = 2^{-7}$

Thus, the expression becomes:

$2^{-7} \times 3^{-3} \times 5^3$

This is simplified as:

$2^{-7} \times 3^{-3} \times 5^3$

So the answer is option A: $2^{-7} \times 3^{-3} \times 5^3$.

Problem 7:

Simplify the following expression:

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

Start by simplifying the powers for each variable:

• For $a$, we have: $a^2 / a^2 = a^0 = 1$
• For $b$, we have: $b^6 / b^{-3} = b^{6+3} = b^9$
• For $c$, we have: $c^6 / c^{-4} = c^{6+4} = c^{10}$

Thus, the simplified expression is:

$b^9c^{10}$

So the answer is $b^9c^{10}$.

Would you like a more detailed explanation of any step?

Here are 5 related questions you might be interested in:

1. How can we simplify powers with negative exponents?
2. What are some common exponent rules for multiplication and division?
3. How do we handle complex expressions with different base powers?
4. What is the process for rationalizing complex expressions?
5. Can you explain more about simplifying expressions involving multiple variables?

Tip: Always check for base similarities when simplifying powers to make the process smoother.