Problem 100 - Elementary Algebra

Question

\(\frac{3\left(4^{2}-5 \cdot 2\right)}{4 \cdot 8-2 \cdot 16}\)

Step-by-Step Solution

Answer

The expression is undefined.

Step 1 - Simplify the Numerator

First, simplify the expression inside the numerator: \[ 4^2 - 5 \cdot 2 \] Calculate the power and the multiplication: \[ 4^2 = 16 \, \text{and} \, 5 \cdot 2 = 10 \] Then subtract: \[ 16 - 10 = 6 \] So the numerator becomes: \[ 3 \cdot 6 \] which equals \[ 18 \].

Step 2 - Simplify the Denominator

Now simplify the expression inside the denominator: \[ 4 \cdot 8 - 2 \cdot 16 \] Calculate the multiplication: \[ 4 \cdot 8 = 32 \, \text{and} \, 2 \cdot 16 = 32 \] Then subtract: \[ 32 - 32 = 0 \] The denominator is 0.

Step 3 - Evaluate the Expression

The simplified fraction is: \[ \frac{18}{0} \] A fraction with 0 in the denominator is undefined.

Key Concepts

Numerator and Denominator

Fractions are made up of two parts: the numerator and the denominator. The numerator is the number above the line and it represents how many parts we have. The denominator is the number below the line, showing the total number of equal parts. For example, in the fraction \(\frac{3}{4}\), 3 is the numerator and 4 is the denominator. This tells us that we have 3 out of 4 parts.

When working on an exercise like \(\frac{3(4^2 - 5 \cdot 2)}{4 \cdot 8 - 2 \cdot 16}\), simplifying both the numerator and the denominator separately is crucial before combining them for the final evaluation.

Let's break down this fraction step by step: For the numerator, we simplify \((4^2 - 5 \cdot 2)\). Similarly, for the denominator, we simplify \((4 \cdot 8 - 2 \cdot 16)\).

Order of Operations

The order of operations is a set of rules that dictate the sequence in which mathematical operations are performed. The commonly used acronym to remember this is PEMDAS:

• P - Parentheses first
• E - Exponents (powers and roots)
• M - Multiplication and
• D - Division (left to right)
• A - Addition and
• S - Subtraction (left to right)

Let's apply these rules to our exercise.

For the numerator, we have: \((4^2 - 5 \cdot 2)\). Follow these steps:

• First, calculate the exponent: \{4^2 = 16\}
• Then, perform multiplication: \{5 \cdot 2 = 10\}
• Finally, subtract: \{16 - 10 = 6\}

Now, multiply by 3, giving us \{3 \cdot 6 = 18\}.

For the denominator, we have: \{4 \cdot 8 - 2 \cdot 16\}. Simplify as follows:

• First, perform multiplications: \{4 \cdot 8 = 32\}
• Then, \{2 \cdot 16 = 32\}
• Finally, subtract: \{32 - 32 = 0\}

Division by Zero

Division by zero is a mathematical concept that occurs when the denominator in a fraction is zero. In mathematics, this is undefined because you can't divide any number by zero. Think of it this way: if you have 18 apples and try to divide them among 0 baskets, it's impossible to distribute apples, just as we can't perform \(\frac{18}{0}\).

In our exercise \(\frac{3(4^2 - 5 \cdot 2)}{4 \cdot 8 - 2 \cdot 16}\), the numerator simplified to 18, and the denominator simplified to 0. Therefore, the expression results in \(\frac{18}{0}\), which is undefined. Always remember, any time you encounter a denominator of zero, the fraction does not have a value. Handle such cases by stating that the expression is undefined.