Problem 113 - College Algebra

Question

Use the order of operations to simplify each expression. \(\frac{5 \cdot 2-3^{2}}{\left[3^{2}-(-2)\right]^{2}}\)

Step-by-Step Solution

Answer

The simplified form of the expression \(\frac{5 \cdot 2 - 3^{2}}{[3^{2} - (-2)]^{2}}\) is \(\frac{1}{121}\).

Step 1: Simplify the Operations Inside the Brackets

In the numerator there's \(5 \cdot 2 - 3^{2}\) and in the denominator there's \([3^{2} - (-2)]^{2}\). Compute the exponentiation first, followed by the multiplication and the subtraction in the numerator. In the denominator, simplify the expression inside the bracket first, then compute the square of the result. The numerator becomes \(5 \cdot 2 - 9=1\) and the denominator becomes \([9 - (-2)]^{2} = [11]^{2} = 121\).

Step 2: Simplify the Entire Expression

With the numerator as 1 and the denominator as 121, the expression becomes \( \frac{1}{121}\), which is the most simplified form of the expression.

Key Concepts

Simplification

Simplification in mathematics refers to the process of breaking down complex expressions into their simplest forms. This is achieved by performing operations such as addition, subtraction, multiplication, and division in a sequence that follows the rules of arithmetic.

When simplifying an expression, it's essential to follow the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division (left to right), Addition and Subtraction (left to right)).

In our exercise, simplification involved carefully dissecting parts of the expression according to these operation rules. We started with the exponentials within the brackets, followed by multiplication and subtraction. Finally, with everything simplified, the complex fraction turns into its simplest form.

Exponents

Exponents are a way to express repeated multiplication of the same number. For instance, in the given expression, the numeral 3 in the numerator is raised to the power of 2, which is written as 3².

This denotes the multiplication of 3 by itself:

• Calculate the power
• 3 x 3 = 9

Exponents are always addressed before other operations unless parentheses dictate otherwise. It is crucial to resolve all exponent terms in an expression early to avoid errors in simplification.

In our example, the denominator required identifying the expression within brackets and calculating 3² before any other changes. Mastering exponents helps in dealing with expressions efficiently and accurately.

Numerator and Denominator

Understanding numerator and denominator roles is key to fraction management in expressions. The numerator is the top part of a fraction, and the denominator is the bottom part. They represent parts of a whole when divided.

In our exercise, the expression's numerator was simplified to 1 through calculating

• 5 × 2 = 10
• subtracting the result of 3², giving 10 - 9 = 1

Meanwhile, for the denominator, complex operations included resolving the bracket expression straight to

• 3² - (-2) which equals 11
• Then, 11² = 121

The complete simplification of both numerator and denominator resulted in the expression \( \frac{1}{121} \). This illustrates how manipulating both parts directly impacts understanding and simplifying fractions.

Expression Evaluation

Expression evaluation involves performing a sequence of operations to find a solution. In our example, we began by looking inside the brackets, which marked priority operations.

Evaluating expressions correctly depends on the rules of order and operation accuracy. If evaluated out of sequence, results could become erroneous.

• Bracket expression and evaluation
• Next, the application of exponents
• Final simplifications

Our finalized result of \( \frac{1}{121} \) highlights the accuracy of expression evaluation. Each mathematical tool and operation complements each other, like distinguishing immediate operations in simplified format and working coherently through complex expressions.

Expression evaluation demands comprehensive understanding of these concepts to ensure the right outcome is consistently achieved.