To simplify the expression \( 16 - 12 + 5 \times 8 + 2^2 / 5^3 - 1 + 4 \), we will follow the order of operations (parentheses, exponents, multiplication and division from left to right, addition and subtraction from left to right).
Let's break it down step by step:
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Calculate the exponentials: \[ 2^2 = 4 \] \[ 5^3 = 125 \]
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Update the expression with these calculations: \[ 16 - 12 + 5 \times 8 + \frac{4}{125} - 1 + 4 \]
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Calculate the multiplication: \[ 5 \times 8 = 40 \]
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Update the expression: \[ 16 - 12 + 40 + \frac{4}{125} - 1 + 4 \]
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Now we can handle addition and subtraction from left to right: \[ 16 - 12 = 4 \] So we update: \[ 4 + 40 + \frac{4}{125} - 1 + 4 \]
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Continue combining terms: \[ 4 + 40 = 44 \] Now: \[ 44 + \frac{4}{125} - 1 + 4 \]
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Then subtract 1: \[ 44 - 1 = 43 \] So we update the expression: \[ 43 + \frac{4}{125} + 4 \]
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Combine with the 4: \[ 43 + 4 = 47 \] Thus, we have: \[ 47 + \frac{4}{125} \]
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Now, combine \( 47 \) and \( \frac{4}{125} \): Convert \( 47 \) into a fraction form: \[ 47 = \frac{47 \times 125}{125} = \frac{5875}{125} \]
Add the two fractions: \[ \frac{5875}{125} + \frac{4}{125} = \frac{5875 + 4}{125} = \frac{5879}{125} \]
Thus, the entire expression simplifies to:
\[ \frac{5879}{125} \]
So, the simplified numerator is 5879.
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