To determine which operator can be inserted to make the expression true, let's define the expression clearly with placeholders for the operators.
The expression is:
\[ -6 + 3 \times 5 ___ \sqrt{16} + 32 \div 8 \]
First, we can resolve the square root and the division:
\(\sqrt{16} = 4\) and \(32 \div 8 = 4\).
Now, substituting these values into the expression, we have:
\[ -6 + 3 \times 5 ___ 4 + 4 \]
Calculating \(3 \times 5 = 15\), we can rewrite it as:
\[ -6 + 15 ___ 4 + 4 \]
Now, simplify the expression components:
Calculating \(-6 + 15 + 4\):
\(-6 + 15 = 9\)
Then,
\(9 + 4 = 13\)
So we have:
\[ 13 ___ 4 \]
Now, we can try different operators to see which makes the expression true.
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Using "+": \[ 13 + 4 = 17 , \text{(not true)} \]
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Using "-": \[ 13 - 4 = 9 , \text{(not true)} \]
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Using "×" (multiplication): \[ 13 \times 4 = 52 , \text{(not true)} \]
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Using "÷" (division): \[ 13 \div 4 = 3.25 , \text{(not true)} \]
Considering these values, we notice that our operations failed to yield equal values, indicating that we might be looking for an equality between two sides of the equation instead.
Thus, we can conclude if none of these combinations yield the desired equality:
The solution that balances both sides will be through a careful adjustment of recognized mathematical rules.
To make the statement true by adjusting operations with values or inserting appropriate key operators might not create a valid straightforward equation. Further context, constraints or equalities should provide clarity.
Please review and confirm, or provide additional details.