Problem 98 - Elementary Algebra

Question

\(\frac{6 \cdot 8-8^{2}}{-4^{2}}\)

Step-by-Step Solution

Answer

-1

Step 1: Calculate the Numerator

First, calculate the expression in the numerator: \(6 \cdot 8 - 8^{2}\). Start by performing the operations inside the numerator separately.

Step 1.1: Multiply 6 and 8

Perform the multiplication: \( 6 \cdot 8 \). \( 6 \cdot 8 = 48 \).

Step 1.2: Square 8

Now, square the number 8: \( 8^2 \). \( 8^2 = 64 \).

Step 1.3: Subtract the Results

Subtract the squared result from the product: \( 48 - 64 \). \( 48 - 64 = -16 \). Therefore, the numerator is -16.

Step 2: Calculate the Denominator

Next, calculate the expression in the denominator: \( -4^2 \). This means squaring -4.

Step 2.1: Square -4

Square the number -4: \( -4^2 \). \( -4^2 = 16 \).

Step 3: Divide the Numerator by the Denominator

Now, divide the result of the numerator by the result of the denominator: \( \frac{-16}{16} \). \( \frac{-16}{16} = -1 \).

Key Concepts

Numerator and Denominator

In algebra, a fraction consists of a numerator and a denominator.
The numerator is the top part which represents how many parts we have.
The denominator is the bottom part that tells us into how many parts the whole is divided. For example, in the fraction \(\frac{a}{b}\), \(a\) is the numerator, and \(b\) is the denominator.

In the given exercise, the numerator is defined by the expression \(6 \cdot 8 - 8^2\). The denominator is the value obtained by squaring -4, i.e., \((-4)^2\). This means:

• Calculate the individual components of the numerator.
• Square the denominator.

Once both values are computed, they can be used to find the value of the fraction.

Multiplication and Squaring

Understanding multiplication and squaring is crucial in simplifying expressions.

• Multiplication involves combining equal groups. For example, \(6 \cdot 8\) essentially means adding 6 groups of 8, which equals 48.
• Squaring a number means multiplying it by itself. For instance, squaring 8 means calculating \(8 \times 8\), leading to 64.

In the numerator of our exercise:
First, we perform the multiplication \(6 \cdot 8\), resulting in 48.
Next, we square the number 8: 64.

Order of Operations

The order of operations is essential for solving algebraic expressions correctly.
It dictates the sequence in which mathematical operations should be performed to avoid errors. The standard order can be remembered using the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)).

• First, handle any operations within parentheses.
• Next, deal with exponents (like squaring).
• Then, perform any multiplication or division as encountered from left to right.
• Finally, any additions or subtractions also from left to right.

In our problem:
1. We start with multiplication and squaring in the numerator.

• Calculate \(6 \cdot 8 = 48\).
• Then \(8^2 = 64\).
• Subtract these results to get the numerator: \(48 - 64 = -16\).

2. Move to the denominator:

• Square -4: \((-4)^2 = 16\).

3. Lastly, divide the results: \(\frac{-16}{16} = -1\).Following this order ensures accurate results.