Mathematical expressions using the number 3 in various arithmetic and algebraic forms.

Actually, you did not upload an image — the content you provided is text, and it appears to be a grid of 15 mathematical expressions involving the number 3, exponents, addition, subtraction, multiplication, and division.

Since there’s no actual “task” or instruction in your message beyond “solve the problem and explain the solution,” I’ll assume the goal is to evaluate each expression and perhaps identify patterns or simplify them.

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Let’s evaluate all 15 expressions one by one. We’ll compute their numerical values step by step.

Recall:
- \(3^3 = 27\)
- Order of operations: Parentheses → Exponents → Multiplication/Division (left to right) → Addition/Subtraction (left to right)

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Row 1

#### 1. \(\frac{3 \times 3 + 3^3}{3}\)
= \(\frac{9 + 27}{3} = \frac{36}{3} = \boxed{12}\)

#### 2. \(3^3 - \frac{3 + 3}{3}\)
= \(27 - \frac{6}{3} = 27 - 2 = \boxed{25}\)

#### 3. \(3 + \left(\frac{3}{3} + 3\right)^3\)
Inside parentheses: \(\frac{3}{3} + 3 = 1 + 3 = 4\)
Then: \(4^3 = 64\)
So: \(3 + 64 = \boxed{67}\)

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Row 2

#### 4. \(\frac{3 \times 3}{3} + 3^3\)
= \(\frac{9}{3} + 27 = 3 + 27 = \boxed{30}\)

#### 5. \(\frac{3}{3} - 3 + 3^3\)
= \(1 - 3 + 27 = -2 + 27 = \boxed{25}\)

#### 6. \(3 + \frac{(3 + 3)^3}{3}\)
First: \(3+3=6\), then \(6^3=216\)
Then: \(\frac{216}{3} = 72\)
So: \(3 + 72 = \boxed{75}\)

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Row 3

#### 7. \(\frac{3 + 3}{3} + 3^3\)
= \(\frac{6}{3} + 27 = 2 + 27 = \boxed{29}\)

#### 8. \(\frac{3 - 3}{3} + 3^3\)
= \(\frac{0}{3} + 27 = 0 + 27 = \boxed{27}\)

#### 9. \(3 + \left(\frac{3 + 3}{3}\right)^3\)
Inside: \(\frac{6}{3} = 2\), then \(2^3 = 8\)
So: \(3 + 8 = \boxed{11}\)

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Row 4

#### 10. \(\frac{3 + 3 \times 3^3}{3}\)
First: \(3 \times 3^3 = 3 \times 27 = 81\)
Then: \(3 + 81 = 84\)
Then: \(\frac{84}{3} = \boxed{28}\)

#### 11. \(\frac{3^3 - 3}{3} + 3\)
Numerator: \(27 - 3 = 24\) → \(\frac{24}{3} = 8\)
Then: \(8 + 3 = \boxed{11}\)

#### 12. \(3 + \left(\frac{3 - 3}{3}\right)^3\)
Inside: \(\frac{0}{3} = 0\), then \(0^3 = 0\)
So: \(3 + 0 = \boxed{3}\)

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Row 5

#### 13. \(\frac{3^3 + 3}{3} + 3\)
Numerator: \(27 + 3 = 30\) → \(\frac{30}{3} = 10\)
Then: \(10 + 3 = \boxed{13}\)

#### 14. \(\frac{3^3 - 3 + 3}{3}\)
Numerator: \(27 - 3 + 3 = 27\) → \(\frac{27}{3} = \boxed{9}\)

#### 15. \(3 + \left(\frac{3}{3}\right)^{3+3}\)
First: \(\frac{3}{3} = 1\), exponent: \(3+3=6\) → \(1^6 = 1\)
So: \(3 + 1 = \boxed{4}\)

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## ✔ Final Answers (in order):

1. 12
2. 25
3. 67
4. 30
5. 25
6. 75
7. 29
8. 27
9. 11
10. 28
11. 11
12. 3
13. 13
14. 9
15. 4

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## 🧠 Observations / Patterns:

- Many expressions involve \(3^3 = 27\), so results often center around that.
- Expressions with \(\frac{3}{3} = 1\) lead to simpler powers (like \(1^6 = 1\)).
- The expression with \((3+3)^3 = 216\) gives the largest value (75).
- Some expressions are identical in result (e.g., #2 and #5 both = 25; #9 and #11 both = 11).

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## 📌 If this was meant to be a puzzle (e.g., find which one equals something specific):

You might need to tell me what the *actual task* is — for example:

> “Which expression equals 27?” → Answer: #8
> “Which one equals 11?” → Answers: #9 and #11
> “Which one is the largest?” → #6 = 75

But since no specific goal was given, I’ve evaluated all 15.

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✔ All 15 expressions have been solved and explained. Let me know if you want to compare them, graph them, or find which ones are equal!