To insert parentheses in the expression \( 5 + 4 \cdot 2 + 6 - 2 \cdot 2 - 1 \) so that it equals 19, we can proceed with the following arrangement:
\[ 5 + (4 \cdot 2) + 6 - (2 \cdot 2) - 1 \]
Now let's calculate the value of this expression step by step:
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Compute \( 4 \cdot 2 \): \[ 4 \cdot 2 = 8 \]
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Compute \( 2 \cdot 2 \): \[ 2 \cdot 2 = 4 \]
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Substitute these values back into the expression: \[ 5 + 8 + 6 - 4 - 1 \]
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Now perform the addition and subtraction in order:
- Start with \( 5 + 8 \): \[ 5 + 8 = 13 \]
- Then add \( 6 \): \[ 13 + 6 = 19 \]
- Next, subtract \( 4 \): \[ 19 - 4 = 15 \]
- Finally, subtract \( 1 \): \[ 15 - 1 = 14 \]
Oh, I realize I misunderstood your requirement. The expression must equal 19. Let's try again by inserting different parentheses.
After examining how to reach a sum of 19, here’s an alternative arrangement:
Revising to: \[ (5 + 4) \cdot 2 + 6 - 2 \cdot 2 - 1 \]
Calculating this gives:
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Calculate \( 5 + 4 \): \[ 5 + 4 = 9 \]
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Now multiply by \( 2 \): \[ 9 \cdot 2 = 18 \]
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Substitute this value back into the expression: \[ 18 + 6 - 4 - 1 \]
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Now perform the addition and subtraction:
- Start with \( 18 + 6 \): \[ 18 + 6 = 24 \]
- Next, subtract \( 4 \): \[ 24 - 4 = 20 \]
- Finally, subtract \( 1 \): \[ 20 - 1 = 19 \]
Thus, the arrangement \( (5 + 4) \cdot 2 + 6 - 2 \cdot 2 - 1 \) indeed evaluates to \( 19 \).