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Let's go through each of these problems one by one:

### (1) \( 138 - 519 - 341 + 655 \)
Perform the operations from left to right:
\[
138 - 519 = -381
\]
\[
-381 - 341 = -722
\]
\[
-722 + 655 = -67
\]
**Answer:** \(-67\)

---

### (2) \( 91 \times 34 \div 13 \)
First, perform the multiplication and then the division:
\[
91 \times 34 = 3094
\]
\[
3094 \div 13 = 238
\]
**Answer:** \(238\)

---

### (3) \( 48.6 + 12.9 \times 3 \)
According to the order of operations (PEMDAS), multiplication comes first:
\[
12.9 \times 3 = 38.7
\]
Now, add:
\[
48.6 + 38.7 = 87.3
\]
**Answer:** \(87.3\)

---

### (4) \( \frac{5}{6} + \frac{1}{3} \times \frac{14}{15} + \frac{7}{9} \)
First, perform the multiplication between fractions:
\[
\frac{1}{3} \times \frac{14}{15} = \frac{14}{45}
\]
Now, add the fractions:
\[
\frac{5}{6} = \frac{75}{90}, \quad \frac{7}{9} = \frac{70}{90}, \quad \frac{14}{45} = \frac{28}{90}
\]
Add all the fractions:
\[
\frac{75}{90} + \frac{28}{90} + \frac{70}{90} = \frac{173}{90}
\]
\[
\frac{173}{90} \text{ is the simplified fraction.}
\]
**Answer:** \(\frac{173}{90}\)

---

### (5) \( 36 - 21 \div 3 + 4 \)
According to the order of operations:
\[
21 \div 3 = 7
\]
Now, subtract and then add:
\[
36 - 7 = 29
\]
\[
29 + 4 = 33
\]
**Answer:** \(33\)

---

### (6) \( 7 \times (1 - 6) - 24 \div (4 - 6) + 23 \)
First, simplify inside the parentheses:
\[
1 - 6 = -5, \quad 4 - 6 = -2
\]
Now, multiply and divide:
\[
7 \times (-5) = -35, \quad 24 \div (-2) = -12
\]
Finally, sum all terms:
\[
-35 - (-12) + 23 = -35 + 12 + 23 = 0
\]
**Answer:** \(0\)

---

### (7) \( (2x + y + 7) \times (3x + 5y) \)
Use the distributive property (FOIL):
\[
= (2x)(3x) + (2x)(5y) + (y)(3x) + (y)(5y) + (7)(3x) + (7)(5y)
\]
\[
= 6x^2 + 10xy + 3xy + 5y^2 + 21x + 35y
\]
Now, combine like terms:
\[
6x^2 + 13xy + 5y^2 + 21x + 35y
\]
**Answer:** \(6x^2 + 13xy + 5y^2 + 21x + 35y\)

---

### (8) \( 2\sqrt{2} + \sqrt{5} \times 2\sqrt{3} \)
First, perform the multiplication:
\[
\sqrt{5} \times 2\sqrt{3} = 2\sqrt{15}
\]
Now, the expression becomes:
\[
2\sqrt{2} + 2\sqrt{15}
\]
**Answer:** \(2\sqrt{2} + 2\sqrt{15}\)

---

Would you like any further clarification on any of these solutions? Here are 5 related questions to explore the concepts further:

1. How do you apply the order of operations (PEMDAS) correctly in more complex expressions?
2. How can fractions be simplified after performing operations like addition or multiplication?
3. What are the steps for distributing terms in polynomial multiplication?
4. How do you rationalize a complex radical expression like \( 2\sqrt{2} + 2\sqrt{15} \)?
5. What is the significance of finding the least common denominator when adding fractions?

**Tip:** Always remember to follow the order of operations (PEMDAS) carefully in multi-step problems to avoid mistakes!

Solve the following math problems: (1) 138 - 519 - 341 + 655, (2) 91 × 34 ÷ 13, (3) 48.6 + 12.9 × 3, (4) 5/6 + 1/3 × 14/15 + 7/9, (5) 36 - 21 ÷ 3 + 4, (6) 7 × (1 - 6) - 24 ÷ (4 - 6) + 23, (7) (2x + y + 7) × (3x + 5y), (8) 2√2 + √5 × 2√3

Explore how to solve multiple types of math problems, including basic arithmetic, fractions, polynomial multiplication, and working with radicals. Each problem breaks down step-by-step solutions for students in grades 8-10.

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