- Evaluate the exponent first, noting that \(-9^2 = -81\).
- Simplify the numerator by adding: \(-81 + 17 = -64\).
- Divide the numerator by the denominator: \(-64 / -4 = 16\).
- The final result is \(\boxed{16}\).
Explanation
- Analyzing the expression and order of operations
The expression to evaluate is \(\frac{-9^2+17}{-4}\). According to the order of operations (often remembered by the acronym PEMDAS or BODMAS), exponentiation must be performed before addition or subtraction. Furthermore, it is important to note that \(-9^2\) is equivalent to \(-(9^2)\), meaning the exponent applies only to the number 9, and the negative sign is applied afterwards. Therefore, we first calculate \(9^2 = 81\), which gives us \(-9^2 = -81\).
- Evaluating the numerator
Now that we have determined the exponent part, we can proceed with the addition in the numerator. Substituting \(-81\) for \(-9^2\), the numerator becomes:
\[-81 + 17 = -64\]
With the numerator simplified, the expression is now:
\[\frac{-64}{-4}\]
- Performing the final division
Finally, we perform the division of the numerator by the denominator. Dividing a negative number by a negative number results in a positive value.
\[\frac{-64}{-4} = 16\]
Thus, the value of the given expression is 16.
Examples
This expression evaluation relates to basic algebraic manipulations encountered in everyday physics and finance. For instance, if you are calculating the net change in a variable over time where a squared factor is subtracted, such as an object's position given by a quadratic motion formula \(s(t) = -t^2 + v_0t + s_0\), understanding order of operations ensures you correctly predict future positions. Mastering these steps helps in correctly modeling real-world trends and budgeting scenarios where specific costs or variables are squared.