Problem 110 - Elementary and Intermediate Algebra

Question

Evaluate each expression. $$ -\left(\frac{8^{2}-10}{2(3)(4)-5(3)}\right) $$

Step-by-Step Solution

Answer

The expression evaluates to -6.

Step 1: Simplifying the Numerator

The given expression is \[-\left(\frac{8^{2}-10}{2(3)(4)-5(3)}\right)\]First, we need to simplify the numerator, which is \[8^2 - 10\]Calculate \(8^2\), which is \(64\). Then subtract \(10\) from \(64\). So, the numerator simplifies to \(64 - 10 = 54\).

Step 2: Simplifying the Denominator

Next, simplify the denominator: \[2(3)(4) - 5(3)\]Calculate the expression \(2 \times 3 \times 4\), which equals \(24\). Then, calculate \(5 \times 3\), which totals \(15\). Now subtract \(15\) from \(24\) to get the denominator value:\(24 - 15 = 9\).

Step 3: Form the Fraction and Divide

Now form the fraction using the simplified values:\[\frac{54}{9}\]Divide \(54\) by \(9\) to simplify the fraction:\[54 \div 9 = 6\].

Step 4: Apply the Negative Sign

The original expression has a negative sign outside the fraction, so we apply it to the simplified result:\[-(6) = -6\].

Key Concepts

Numerical Simplification

Numerical simplification involves taking complex mathematical expressions and making them simpler and easier to handle.
This is a foundational skill in algebra that allows you to break down and resolve equations efficiently.In the given expression, we needed to simplify both the numerator and the denominator. For the numerator, \(8^2 - 10\) simplifies to \(54\) because \(8^2\) equals \(64\) and subtracting \(10\) gives us \(54\).
Recognizing powers, like \(8^2\), is crucial, as it involves multiplying the base number by itself. Here, calculating powers gives a precise numerical result, which is then used in further operations.The denominator's computation, \(2(3)(4) - 5(3)\), also shows numerical simplification at work.

• \(2 \times 3 \times 4\) results in \(24\)
• \(5 \times 3\) results in \(15\)

By subtracting these two products (\(24 - 15\)), you simplify the denominator to \(9\). Simplification processes make complex arithmetic steps much more manageable and clear.

Fraction Operations

Working with fractions means understanding how to manipulate the numerator and the denominator effectively.
In our exercise, after simplifying both parts, we are left with a fraction \(\frac{54}{9}\).Dividing the numerator by the denominator is a key operation here, showing the essence of dealing with fractions.
By calculating \(54 \div 9\), you obtain \(6\). This process of dividing is crucial for testifying how fractions can be simplified into whole numbers or smaller fractions when possible.The entire simplification of the expression becomes a task of converting a complex part over another into its simplest form.

• Make sure you correctly align and simplify individual parts.
• Perform basic arithmetic operations checking your steps to confirm the accuracy of the solution.

Understanding these fraction operations helps in reaching the concluding answer efficiently.

Order of Operations

To solve any algebraic expression correctly, one must follow the order of operations.
This refers to the specific sequence you execute different arithmetic operations: Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right) - commonly remembered by the acronym PEMDAS. In our exercise, we followed this rule by first tackling the calculation within the parentheses (denominator), then the exponentiation (in the numerator), and finally the subtraction and multiplication operations.
Once both the numerator and the denominator were simplified separately by their order, we moved on to fraction operations. Proper adherence to the order of operations prevents mathematical errors and ensures that expressions are simplified correctly.

• Start solving inside the parentheses first.
• Take care of exponents before multiplication and division.
• Finish with any addition or subtraction.

Mastering this concept is crucial for solving not only simplified problems but also advanced algebraic challenges.