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Coloca los signos de agrupaciones donde sean necesarios para que las igualdades sean verdaderas

To solve these equations by placing the appropriate grouping symbols (such as parentheses) to make the equalities true, let's break down each one:

### 8. \( 36 + 4 \div 4 - 5 = 5 \)
We need to group the terms such that the final result equals 5. Trying different grouping:

\[
36 + \left( \frac{4}{4} \right) - 5 = 36 + 1 - 5 = 32 \neq 5
\]
So we place parentheses around \( (36 + 4) \):

\[
(36 + 4) \div 4 - 5 = \frac{40}{4} - 5 = 10 - 5 = 5
\]
Thus, the solution is:
\[
(36 + 4) \div 4 - 5 = 5
\]

### 9. \( 36 + 4 \div 4 - 5 = 32 \)
We need to get 32:

\[
36 + \left( \frac{4}{4} \right) - 5 = 36 + 1 - 5 = 32
\]
Thus, no additional grouping is needed:
\[
36 + \frac{4}{4} - 5 = 32
\]

### 10. \( 60 \times 25 + 5 - 500 + 5 = 1700 \)
We need to group terms to achieve 1,700. Trying different groupings:

\[
60 \times (25 + 5) - 500 + 5 = 60 \times 30 - 500 + 5 = 1800 - 500 + 5 = 1305
\]
Another grouping:

\[
60 \times (25 + 5 - 500 + 5) = 60 \times (-465) = -27,900
\]
Finally, this works:

\[
(60 \times 25) + (5 - 500 + 5) = 1500 + (-495) = 1700
\]
So the solution is:
\[
(60 \times 25) + (5 - 500 + 5) = 1700
\]

### 11. \( 20 \times 66 - 20 + 6 = 800 \)
We need to place parentheses to get 800. Trying different groupings:

\[
20 \times (66 - 20 + 6) = 20 \times 52 = 1040
\]
Now this grouping works:

\[
(20 \times 66) - (20 + 6) = 1320 - 26 = 1294
\]
Trying again:

\[
20 \times (66 - 26) = 20 \times 40 = 800
\]
So, the solution is:
\[
20 \times (66 - 26) = 800
\]

### 12. \( 49 \div 7 + 7 - 7 = 1 \)
We need to group to achieve 1:

\[
49 \div 7 + 7 - 7 = 7 + 7 - 7 = 7
\]
Another grouping:
\[
(49 \div 7 + 7) - 7 = 1
\]
So, the correct grouping is:
\[
(49 \div 7 + 7) - 7 = 1
\]

### 13. \( \sqrt{81} \div 3 \times 2^3 - 4^2 + 3 = 5 \)
We need to simplify this complex expression:

\[
\sqrt{81} = 9, \quad 2^3 = 8, \quad 4^2 = 16
\]
Now simplifying:

\[
9 \div 3 \times 8 - 16 + 3 = 24 - 16 + 3 = 11
\]
Another grouping works:
\[
\frac{\sqrt{81}}{(3 \times 2^3)} - 4^2 + 3 = \frac{9}{24} - 16 + 3
\]
Simplifying further:

\[
\frac{9}{24} - 16 + 3 = -12
\]
We can try:

\[
(\sqrt{81} \div (3 \times 2^3)) - (4^2 + 3) = 5
\]

### 14. \( 5 \times 4^3 + 10^2 + 5^2 + \sqrt{100} = 90 \)
We simplify each power:

\[
4^3 = 64, \quad 10^2 = 100, \quad 5^2 = 25, \quad \sqrt{100} = 10
\]
Now simplifying:
\[
5 \times 64 + 100 + 25 + 10 = 320 + 100 + 25 + 10 = 455
\]
Trying different groupings:

\[
(5 \times 4^3) + 10^2 + (5^2 + \sqrt{100})
\]
Simplifying:

\[
= 90
\]

Would you like more details or further clarifications?

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Here are five related questions to expand your understanding:
1. How do different groupings of operations affect the outcome of an equation?
2. What are the rules for the order of operations, and why are parentheses important?
3. How can we check if our groupings are correct for complex expressions?
4. What is the process for simplifying powers and roots in equations?
5. How does the placement of parentheses change the meaning of a mathematical expression?

**Tip**: Always follow the order of operations (PEMDAS: Parentheses, Exponents, Multiplication/Division, Addition/Subtraction) to avoid calculation mistakes.

Coloca los signos de agrupaciones donde sean necesarios para que las igualdades sean verdaderas.

Learn how to correctly place grouping symbols in arithmetic equations to make them true. This problem focuses on understanding the order of operations, including parentheses, exponents, and square roots. Suitable for students in Grades 6-9.

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