Use the rule for order of operations to simplify the following. Leave your answer as a proper fraction or a mixed number (7/9 + 3/8)(1/6 + 4/5)

Let's break down the given expression \((\frac{7}{9} + \frac{3}{8})(\frac{1}{6} + \frac{4}{5})\) step by step using the order of operations.

### Step 1: Simplify the terms inside the parentheses.

1. Add \(\frac{7}{9} + \frac{3}{8}\):
   - The least common denominator (LCD) of 9 and 8 is 72.
   - Rewrite the fractions with denominator 72:
     \[
     \frac{7}{9} = \frac{7 \times 8}{9 \times 8} = \frac{56}{72}, \quad \frac{3}{8} = \frac{3 \times 9}{8 \times 9} = \frac{27}{72}
     \]
   - Add the two fractions:
     \[
     \frac{56}{72} + \frac{27}{72} = \frac{56 + 27}{72} = \frac{83}{72}
     \]

2. Add \(\frac{1}{6} + \frac{4}{5}\):
   - The least common denominator (LCD) of 6 and 5 is 30.
   - Rewrite the fractions with denominator 30:
     \[
     \frac{1}{6} = \frac{1 \times 5}{6 \times 5} = \frac{5}{30}, \quad \frac{4}{5} = \frac{4 \times 6}{5 \times 6} = \frac{24}{30}
     \]
   - Add the two fractions:
     \[
     \frac{5}{30} + \frac{24}{30} = \frac{5 + 24}{30} = \frac{29}{30}
     \]

### Step 2: Multiply the results from each pair of parentheses.

Now multiply \(\frac{83}{72}\) by \(\frac{29}{30}\):
\[
\frac{83}{72} \times \frac{29}{30} = \frac{83 \times 29}{72 \times 30} = \frac{2407}{2160}
\]

### Step 3: Simplify the result if possible.

The fraction \(\frac{2407}{2160}\) cannot be simplified further, since 2407 and 2160 have no common factors.

Thus, the simplified answer is:
\[
\boxed{\frac{2407}{2160}}
\]

Would you like further clarification or details on any steps?

Here are 5 related questions to explore further:
1. How do you find the least common denominator (LCD) of two fractions?
2. What is the process for converting an improper fraction to a mixed number?
3. Can you explain how to multiply fractions step by step?
4. How do you simplify fractions after multiplication?
5. What are the key rules for the order of operations in arithmetic?

**Tip:** When adding or subtracting fractions, always look for the least common denominator (LCD) to make the calculation simpler!

Use the rule for order of operations to simplify the following. Leave your answer as a proper fraction or a mixed number: (7/9 + 3/8)(1/6 + 4/5)

Learn how to simplify the expression (7/9 + 3/8)(1/6 + 4/5) using the order of operations. This detailed guide walks through adding and multiplying fractions step-by-step, with tips on finding the least common denominator (LCD). Perfect for Grades 6-8.

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