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To solve this problem, we need to place parentheses in the expression so that the calculation results in 36. Let's examine the provided options:

The given expression is:

\[
2^5 + 31 \times \frac{2^2}{\sqrt{16}} + 3^3 = 36
\]

Let's evaluate each option:

1. **Option 1:**
   \[
   \frac{2 \cdot (5 + 31 \cdot 2^2)}{\sqrt{16}} + 3^3
   \]
   This option places parentheses around \( 5 + 31 \cdot 2^2 \). The calculation is:
   \[
   5 + 31 \times 4 = 5 + 124 = 129
   \]
   Then, multiply by 2:
   \[
   2 \times 129 = 258
   \]
   Divide by \(\sqrt{16} = 4\):
   \[
   \frac{258}{4} = 64.5
   \]
   Finally, add \(3^3 = 27\):
   \[
   64.5 + 27 = 91.5 \quad \text{(not equal to 36)}
   \]

2. **Option 2:**
   \[
   2^5 + 31 \cdot \frac{2^2}{\sqrt{16} + 3^3}
   \]
   Here, the parentheses are around \(\sqrt{16} + 3^3\):
   \[
   \sqrt{16} = 4 \quad \text{and} \quad 3^3 = 27
   \]
   So, the denominator becomes:
   \[
   4 + 27 = 31
   \]
   The expression becomes:
   \[
   2^5 + 31 \times \frac{4}{31} = 32 + 4 = 36
   \]
   This calculation gives the correct result of 36.

3. **Option 3:**
   \[
   \frac{(2^5 + 31) \cdot 2^2}{\sqrt{16}} + 3^3
   \]
   Similar to the first option, but the parenthesis includes \(2^5 + 31\):
   \[
   32 + 31 = 63
   \]
   Multiply by \(4\):
   \[
   63 \times 4 = 252
   \]
   Then divide by \(\sqrt{16} = 4\):
   \[
   \frac{252}{4} = 63
   \]
   Finally, add \(27\):
   \[
   63 + 27 = 90 \quad \text{(not equal to 36)}
   \]

4. **Option 4:**
   \[
   2^5 + 31 \times \frac{2^2}{\sqrt{(16 + 3^3)}}
   \]
   This option places parentheses around \(16 + 3^3\):
   \[
   16 + 27 = 43
   \]
   Then take the square root:
   \[
   \sqrt{43} \quad \text{(which is not an integer and won't simplify to give 36)}
   \]

**Correct Answer:** Option 2 is the correct placement of parentheses to make the expression equal to 36.

Do you want more details or have any questions?

Here are five related questions:
1. How does order of operations affect the placement of parentheses?
2. What are some general strategies for simplifying complex algebraic expressions?
3. Can parentheses change the outcome of arithmetic expressions in any context?
4. How does the distributive property interact with parentheses in expressions?
5. What are other common uses of parentheses in mathematical expressions?

**Tip:** When simplifying expressions, always perform operations inside parentheses first according to the order of operations (PEMDAS/BODMAS).

Make the following calculation correct by placing a parenthesis: 2^5 + 31 × 2^2 / √16 + 3^3 = 36.

Learn how to correctly place parentheses in the expression 2^5 + 31 × 2^2 / √16 + 3^3 = 36. This problem involves key algebraic concepts like exponents, square roots, and the order of operations. Suitable for Grades 7-9.

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