To evaluate the expression \(10.6 + 3.3^2 + 610 \cdot 6 + 3 \cdot 3^2 + 6\), we need to do the calculations step by step.
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Calculate \(3.3^2\): \[ 3.3^2 = 3.3 \times 3.3 = 10.89 \]
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Calculate \(3 \cdot 3^2\): \[ 3^2 = 9 \quad \text{so} \quad 3 \cdot 3^2 = 3 \cdot 9 = 27 \]
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Calculate \(610 \cdot 6\): \[ 610 \cdot 6 = 3660 \]
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Now, sum everything together: \[ 10.6 + 10.89 + 3660 + 27 + 6 \]
Let's add these values step by step:
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First, add \(10.6 + 10.89\): \[ 10.6 + 10.89 = 21.49 \]
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Next, add \(21.49 + 3660\): \[ 21.49 + 3660 = 3681.49 \]
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Then, add \(3681.49 + 27\): \[ 3681.49 + 27 = 3708.49 \]
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Finally, add \(3708.49 + 6\): \[ 3708.49 + 6 = 3714.49 \]
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So, the value of the expression \(10.6 + 3.3^2 + 610 \cdot 6 + 3 \cdot 3^2 + 6\) is:
\[ \boxed{3714.49} \]