Let's break down the expression step by step:

The given expression is:

$\frac{7^2 - 6 \cdot 7 + 9}{9 - 7^3} + \frac{4 \cdot 7 + 8}{7 + 3}$

Step 1: Simplify the first part of the expression

• $7^2 = 49$
• $6 \cdot 7 = 42$
• $9 - 7^3$: First calculate $7^3 = 343$, so the denominator becomes $9 - 343 = -334$.

Now the first fraction becomes:

$\frac{49 - 42 + 9}{-334} = \frac{16}{-334}$

Step 2: Simplify the second part of the expression

• $4 \cdot 7 = 28$
• $7 + 3 = 10$

Now the second fraction becomes:

$\frac{28 + 8}{10} = \frac{36}{10} = 3.6$

Step 3: Combine both parts

Now, combining both fractions:

$\frac{16}{-334} + 3.6$

We can express $\frac{16}{-334}$ as approximately $-0.0479$.

Thus:

$-0.0479 + 3.6 \approx 3.5521$

Final Answer:

$3.5521$

Would you like further details on any part of this process?

Related Questions:

1. How do you handle exponents and powers in algebraic expressions?
2. What is the significance of simplifying fractions before combining terms?
3. Can negative exponents change the behavior of a fraction significantly?
4. What are the rules for operations on fractions with negative denominators?
5. How do you decide when to approximate the result for practical purposes?

Tip: Always double-check the signs when working with powers and subtraction; small mistakes can lead to larger errors!