Consider this expression:
$\left|m^2-7\right|+n^2$
When $m=-2$ and $n=5$, what is the value of the expression?
We are given the expression \(\left|m^2 - 7\right| + n^2\) and the values \(m = -2\) and \(n = 5\). Our task is to substitute these values into the expression and evaluate it step by step, keeping in mind the properties of the absolute value function. The absolute value of a number \(x\), denoted \(\left|x\right|\), is its distance from zero on the number line, which is equal to \(x\) if \(x \ge 0\) and \(-x\) if \(x < 0\).
First, we substitute \(m = -2\) into the squared term \(m^2\):
\[m^2 = (-2)^2 = 4.\]
Next, we subtract 7 from this result:
\[m^2 - 7 = 4 - 7 = -3.\]
Now, we apply the absolute value function:
\[\left|m^2 - 7\right| = \left|-3\right| = 3.\]
Next, we substitute \(n = 5\) into the term \(n^2\):
\[n^2 = 5^2 = 25.\]
Finally, we sum the absolute value term and the square term to find the total value of the expression:
\[\left|m^2 - 7\right| + n^2 = 3 + 25 = 28.\]
Thus, the value of the expression for \(m = -2\) and \(n = 5\) is \(\boxed{28}\).
Absolute value calculations are fundamental in many areas, such as physics, finance, and data analysis. For example, if you are tracking the temperature change or the difference between your current account balance and a target savings goal, you often deal with deviations. The expression \(|m^2 - 7| + n^2\) could represent a simplified model for the total error or cost associated with a process where \(m\) represents a variable that causes squared fluctuations and \(n\) represents a separate, constant-influenced scaling factor. Understanding how to compute these values ensures you can accurately interpret such models.