Verified Answer :

  • Substitute \(a = -4\) into the first term to get \((-4)^2 = 16\).
  • Substitute \(b = 6\) into the second term to get \(\frac{12}{6} = 2\).
  • Add the calculated components together: \(16 + 2 = 18\).
  • The final result is \(\boxed{18}\).

Explanation

  1. Problem Analysis and Data Substitution

The objective is to evaluate the expression \(a^2 + \frac{12}{b}\) given the specific values \(a = -4\) and \(b = 6\). This problem requires basic substitution followed by standard arithmetic operations, including exponentiation, division, and addition. By substituting these values into the expression, we can reduce the algebraic form to a single numerical value.

  1. Calculating Individual Terms

First, substitute \(a = -4\) into the term \(a^2\). Squaring a negative number results in a positive value:
\[(-4)^2 = (-4) \times (-4) = 16.\]
Next, substitute \(b = 6\) into the term \(\frac{12}{b}\). Performing the division, we get:
\[\frac{12}{6} = 2.\]

  1. Final Calculation

Finally, add the results obtained from the previous steps to find the value of the full expression:
\[a^2 + \frac{12}{b} = 16 + 2 = 18.\]
Thus, evaluating the expression with the given values yields a final result of 18.

Examples

Consider a scenario where you are calculating the total power output of a system. Let the power consumption of one component be represented by the square of the voltage deviation, \(a^2\), and another component be represented by \(\frac{12}{b}\), where \(b\) is a resistance value. If \(a = -4\) volts and \(b = 6\) ohms, calculating the total power gives the total energy efficiency requirement for that specific state. Using algebraic substitution helps engineers quickly assess performance under varying operating conditions.

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