Consider the expression: (16-12)^2+5(8+2^2)/5^3-(1+4)^2. What is the simplified numerator?

4^2 + 60

76

100

54

1 answer

Let's simplify the expression step-by-step:

  1. Start with \( (16 - 12)^2 + 5(8 + 2^2) / 5^3 - (1 + 4)^2 \).

  2. Calculate each part:

    • \( (16 - 12) = 4 \) so \( (16 - 12)^2 = 4^2 = 16 \).
    • Calculate \( 2^2 = 4 \), hence \( 8 + 2^2 = 8 + 4 = 12 \).
    • Therefore, \( 5(8 + 2^2) = 5 \cdot 12 = 60 \).
    • We also need \( 5^3 = 125 \), so \( \frac{60}{125} = \frac{60}{125} = \frac{12}{25} \).
    • Lastly, \( (1 + 4) = 5 \), so \( (1 + 4)^2 = 5^2 = 25 \).
  3. Now substitute back into the expression: \[ 16 + \frac{12}{25} - 25 \]

  4. To proceed with addition and subtraction, convert \( 16 \) and \( 25 \) into fractions with a common denominator of 25:

    • \( 16 = \frac{16 \times 25}{25} = \frac{400}{25} \).
    • \( 25 = \frac{625}{25} \).
  5. The expression now looks like this: \[ \frac{400}{25} + \frac{12}{25} - \frac{625}{25} \]

  6. Combining these fractions: \[ \frac{400 + 12 - 625}{25} = \frac{412 - 625}{25} = \frac{-213}{25} \]

Thus, the simplified numerator is \(-213\). However, as the options provided are positive integers, it seems like we should look at the problem differently, particularly the numerator \(4^2 + 60\) provided in the options.

Let's evaluate that expression:

  • \(4^2 = 16\)
  • So, \(4^2 + 60 = 16 + 60 = 76\)

Lastly, the simplified numerator in the context of that question leads us to the option: 76