Let's simplify the given expression step by step.

The expression is:

$\frac{8 + 9(1 + 7)}{7^2 + 7}$

Step 1: Simplify the numerator

In the numerator: $8 + 9(1 + 7)$ Simplify $1 + 7$ first: $8 + 9 \times 8$ Multiply $9 \times 8$: $8 + 72 = 80$ So, the numerator is $80$.

Step 2: Simplify the denominator

In the denominator: $7^2 + 7$ Calculate $7^2$ first: $49 + 7 = 56$ So, the denominator is $56$.

Step 3: Write the simplified fraction

The fraction is now: $\frac{80}{56}$

Step 4: Simplify the fraction

Simplify $\frac{80}{56}$ by dividing both numerator and denominator by their greatest common divisor (GCD), which is $8$: $\frac{80 \div 8}{56 \div 8} = \frac{10}{7}$

Final Answer:

The simplified expression is: $\frac{10}{7}$

Would you like further clarification or additional examples? Here are some follow-up questions to expand your understanding:

1. What is the general process for simplifying fractions?
2. How do you find the greatest common divisor (GCD) of two numbers?
3. What other operations or transformations could be applied to simplify similar expressions?
4. How would this expression change if additional terms were added to the numerator or denominator?
5. How does understanding the order of operations (PEMDAS) help in solving these problems?

Tip: Always simplify inside parentheses and exponents before proceeding to multiplication, division, addition, or subtraction!