Problem 99 - Elementary Algebra

Question

\(\frac{4\left(5^{2}-6 \cdot 4\right)}{3 \cdot 4-6 \cdot 2}\)

Step-by-Step Solution

Answer

The expression is undefined due to division by zero.

Step 1 - Simplify the Numerator Inside the Parentheses

Start by simplifying the expression inside the parentheses in the numerator: \(5^2 - 6 \times 4\). Calculate \(5^2\) first: \(5^2 = 25\). Next, calculate the product of \(6\) and \(4\): \(6 \times 4 = 24\). Now, subtract the result from the exponentiation: \(25 - 24 = 1\).

Step 2 - Simplify the Modified Numerator

Now multiply \(4\) by the result from Step 1: \(4 \times 1 = 4\).

Step 3 - Simplify the Denominator

Simplify the expression inside the denominator: \(3 \times 4 - 6 \times 2\). First, calculate \(3 \times 4\): \(3 \times 4 = 12\). Next, calculate the product of \(6\) and \(2\): \(6 \times 2 = 12\). Now, subtract the products: \(12 - 12 = 0\).

Step 4 - Divide the Numerator by the Denominator

Now, try to divide the numerator by the denominator: \(\frac{4}{0}\). Any division by zero is undefined in mathematics.

Key Concepts

Numerator and Denominator

In an algebraic fraction, the **numerator** is the expression above the line, and the **denominator** is the expression below the line. They are key parts of any fraction. Let's look at the fraction: \(\frac{4(5^2-6·4)}{3·4-6·2}\). \(
\) The formula can be divided into:
\(
\)- Numerator: 4(5^2 - 6·4)
- Denominator: 3·4 - 6·2
Understanding how to manipulate these parts separately is crucial when solving algebra problems. First, simplify the numerator by performing the operations within the parentheses. Then, simplify the denominator in a similar fashion. Once simplified, the next step typically involves dividing those two results. If either part simplifies to zero, special rules apply, as we’ll see in another section.

Order of Operations

When simplifying algebraic expressions, follow the **order of operations**, often remembered by the acronym PEMDAS: Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).
In our exercise, start with the parentheses inside the numerator and denominator. For the numerator, \(5^2 - 6 \times 4\) breaks down as follows:

• Calculate the exponent: \(5^2 = 25\)
• Perform the multiplication: \(6 \times 4 = 24\)
• Subtract: \(25 - 24 = 1\)

Integrate and multiply by 4 outside the parentheses: \(4 \times 1 = 4\)
For the denominator, the steps are:

• Multiply: \(3 \times 4 = 12\)
• Multiply: \(6 \times 2 = 12\)
• Subtract: \(12 - 12 = 0\)

Following these steps precisely ensures accuracy and avoids unnecessary mistakes.

Undefined Expressions

Some expressions, when simplified, can lead to results that are undefined in mathematics, typically due to division by zero. An **undefined expression** occurs in our example when we try to divide 4 by the denominator which simplifies to zero:\(\frac{4}{0}\).
This is because any number divided by zero is not defined: it does not yield a real number solution. For instance, in our exercise, the final step attempts to divide 4 by 0, leading to an undefined expression. Hence, whenever solving equations, if you encounter 0 in the denominator, stop there. That fraction does not have a defined value. Understanding this concept is essential to avoid critical errors in mathematical problem-solving.