Simplify the following: [tex]$\frac{|-7-(-4)|-|-5|}{-5^2}$[/tex]
Answer (2)
Simplify inside the absolute value: − 7 − ( − 4 ) = − 3 .
Evaluate the absolute values: ∣ − 3∣ = 3 and ∣ − 5∣ = 5 .
Simplify the numerator: 3 − 5 = − 2 .
Simplify the denominator: − 5 2 = − 25 . The simplified fraction is 25 2 . Therefore, the numerator is 2 and the denominator is 25 .
Explanation
Understanding the Problem We are asked to simplify the expression − 5 2 ∣ − 7 − ( − 4 ) ∣ − ∣ − 5∣ and find the simplified numerator and denominator.
Simplifying Inside Absolute Value First, let's simplify the expression inside the absolute value in the numerator: − 7 − ( − 4 ) = − 7 + 4 = − 3 .
Evaluating Absolute Values Next, we evaluate the absolute values in the numerator: ∣ − 3∣ = 3 and ∣ − 5∣ = 5 .
Simplifying the Numerator Now, we simplify the numerator: ∣ − 7 − ( − 4 ) ∣ − ∣ − 5∣ = ∣ − 3∣ − ∣ − 5∣ = 3 − 5 = − 2 .
Simplifying the Denominator Then, we simplify the denominator: − 5 2 = − ( 5 × 5 ) = − 25 . Note that the exponent applies only to the 5, not to the negative sign.
Writing the Simplified Fraction So, the simplified fraction is − 25 − 2 = 25 2 . Therefore, the numerator is 2 and the denominator is 25.
Final Answer The simplified numerator is 2 and the simplified denominator is 25.
Examples
Understanding how to simplify expressions with absolute values and exponents is crucial in various fields, such as physics and engineering, where complex calculations are common. For instance, when calculating the magnitude of a force vector or analyzing electrical circuits, you often encounter expressions involving absolute values and exponents. Simplifying these expressions accurately helps in obtaining precise results and making informed decisions. This skill is also fundamental in computer science for optimizing algorithms and data structures.
To simplify − 5 2 ∣ − 7 − ( − 4 ) ∣ − ∣ − 5∣ , first calculate the absolute values to get a numerator of − 2 and a denominator of − 25 . The final simplified fraction becomes 25 2 .
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