Type the correct answer in the box. Use numerals instead of words. Consider this expression: [tex]$\left|m^2+n^2\right|$[/tex] When [tex]$m=-5$[/tex] and [tex]$n=3$[/tex], the value of the expression is $\boxed{}$ .
Answer (2)
Substitute the given values m = − 5 and n = 3 into the expression m 2 + n 2 .
Calculate the squares: ( − 5 ) 2 = 25 and ( 3 ) 2 = 9 .
Add the squared values: 25 + 9 = 34 .
Take the absolute value: ∣ 34 ∣ = 34 . The value of the expression is 34 .
Explanation
Understanding the Problem We are given the expression m 2 + n 2 and the values m = − 5 and n = 3 . Our goal is to find the value of the expression when we substitute these values for m and n .
Substitution First, we substitute m = − 5 and n = 3 into the expression: ( − 5 ) 2 + ( 3 ) 2
Calculating Squares Next, we calculate the squares of m and n :
( − 5 ) 2 = ( − 5 ) × ( − 5 ) = 25 3 2 = 3 × 3 = 9
Substituting Squared Values Now, we substitute these values back into the expression: ∣ 25 + 9 ∣
Addition We add the numbers inside the absolute value: 25 + 9 = 34 So the expression becomes: ∣ 34 ∣
Absolute Value Finally, we take the absolute value of 34. Since 34 is a positive number, its absolute value is simply 34: ∣ 34 ∣ = 34
Examples
Absolute value expressions are used in many real-world applications, such as calculating distances or errors. For example, if you are measuring the distance between two points, you might get a negative value due to the direction, but the actual distance is always positive. Similarly, in engineering, the absolute value is used to determine the magnitude of a deviation from a target value, regardless of whether the deviation is positive or negative. Understanding how to evaluate expressions with absolute values is crucial for solving problems in physics, engineering, and computer science.
The value of the expression m 2 + n 2 when m = − 5 and n = 3 is 34. This is calculated by substituting the values, finding the squares, and taking the absolute value. Therefore, the answer is 34 .
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