Problem 119 - College Algebra

Question

Use the order of operations to simplify each expression. \(\frac{(5-6)^{2}-2|3-7|}{89-3 \cdot 5^{2}}\)

Step-by-Step Solution

Answer

\(-0.5\)

Step 1: Simplify the Expressions Inside the Brackets and Absolute Value

Start by simplifying the expressions \( (5-6)^{2} \) and \( |3-7| \). Calculate \( 5-6 \) to get -1, and \( |3-7| \) is equal to 4 because the absolute value of a number refers to its distance from zero on the number line, disregarding the direction. Hence, \( |-4| = 4 \). The expression now becomes \( \frac{(-1)^{2}- 2 \cdot 4}{89-3 \cdot 5^{2}} \)

Step 2: Compute Powers and Multiplication

Next, evaluate exponents, i.e., compute \( (-1)^2 \) to get 1, and perform multiplication \( 2 \cdot 4 \) which equals 8. Also, multiply \( 3 \cdot 5^{2} \) which results in 75. Substitute these in the expression as follows \( \frac{1 - 8}{89 - 75} \)

Step 3: Perform Addition and Subtraction

Continue by performing additions and subtractions. So, \( 1 - 8 = -7 \), and \( 89 - 75 = 14 \). Hence, the expression now becomes \( \frac{-7}{14} \)

Step 4: Division

Finally, divide -7 by 14 to get the resulting value of the expression. This gives \( -0.5 \)

Key Concepts

Simplifying Expressions

When simplifying expressions, it's important to follow the order of operations: parentheses, exponents, multiplication and division (from left to right), and finally addition and subtraction (PEMDAS). This method ensures that expressions are simplified correctly.
In our expression, we first looked inside the brackets. The term \((5-6)^2\) showed us \(-1\), and when squared, it became 1. Similarly, the absolute value \(|3-7|\) helped us understand that we are dealing with a positive distance from zero, leading to a result of 4. By calculating these components step by step, we simplify the initial complex expression.

Absolute Value

The absolute value of a number simply measures how far the number is from zero on a number line, regardless of direction. This concept is represented by two vertical bars, like this: \(|x|\).
For example, in our expression \(|3-7|\), we calculate the absolute value of -4, which is 4. It is crucial to remember that the absolute value always produces a non-negative result. This property ensures that we consistently measure magnitude without considering direction. Understanding absolute value helps us highlight the true size of a number, paving the way to simplifying expressions accurately.

Exponents

Exponents are a shorthand to express repeated multiplication. For instance, \(5^2\) means multiplying 5 by itself, giving us 25. In our problem, handling \((-1)^2\), we took \(-1\) and multiplied it by itself, resulting in 1.
It's important to apply exponents carefully, observing any signs associated with base numbers. Multiplying a negative number by itself an even number of times leads to a positive result, while an odd number of times results in a negative result. Mastering exponents allows us to navigate expressions effortlessly and powerfully.

Fractions in Algebra

Fractions in algebra represent division operations within expressions, composed of a numerator and a denominator. In our exercise, the expression becomes \(\frac{-7}{14}\) after simplifying all components.
Fractions simplify by dividing both the numerator and the denominator by their greatest common factor (GCF). In this case, dividing both -7 and 14 by 7 results in \(-\frac{1}{2}\) or -0.5. Understanding how fractions work, including simplification and division, is essential in solving algebraic expressions smoothly and accurately. It forms the basis for more complex algebraic concepts.