Problem 74 - Elementary Algebra

Question

For exercises 15-100, evaluate. $$ \frac{-8-8^{2}}{(3)(-2)} $$

Step-by-Step Solution

Answer

12

Step 1: Simplify the numerator

Calculate the value inside the numerator. Evaluate \(8^2\) first: \(8^2 = 64\). Then subtract 64 from -8: \( -8 - 64 = -72 \). So, the numerator simplifies to -72.

Step 2: Simplify the denominator

Multiply the values in the denominator: \(3 \times -2 = -6\). So, the denominator simplifies to -6.

Step 3: Divide the simplified values

Now divide the simplified numerator by the simplified denominator: \(\frac{-72}{-6} = 12\).

Key Concepts

Simplifying Fractions

Simplifying fractions is an essential skill in algebra. Before you can simplify a fraction, both the numerator and the denominator must be as simple as possible.

In our example, the fraction is \(\frac{-8-8^{2}}{(3)(-2)}\). First, we need to simplify both the numerator and the denominator separately.

For the numerator, follow these steps:

• Calculate the exponent: \(8^2 = 64\).
  
• Subtract 64 from -8: \(-8 - 64 = -72\).
  

Now that the numerator is simplified to -72, we move on to the denominator:

• Multiply the values in the parentheses: \(3 \times -2 = -6\).
  

So, the problem now reads \(\frac{-72}{-6}\). The last step is to divide these two simplified values:

• Divide: \(\frac{-72}{-6} = 12\).
  

By simplifying fractions, you make expressions easier to work with and often reveal more intuitive results.

Order of Operations

Understanding the order of operations is crucial for evaluating algebraic expressions accurately. It's commonly remembered by the acronym PEMDAS, which stands for:

• P: Parentheses
  
• E: Exponents
  
• M: Multiplication
  
• D: Division
  
• A: Addition
  
• S: Subtraction
  

Always follow this order to ensure you solve algebraic expressions correctly.

For our expression \(\frac{-8-8^{2}}{(3)(-2)}\), follow PEMDAS:

• First, solve the exponent \(8^2 = 64\).
  
• Then, handle any parentheses or grouping: Multiply what's inside the parentheses in the denominator: \(3 \times -2 = -6\).
  
• Next, perform the subtraction in the numerator: \-8 - 64 = -72\.
  
• Finally, divide the simplified numerator by the simplified denominator: \(\frac{-72}{-6} = 12\).
  

Paying attention to the order of operations prevents mistakes and ensures you find the correct answer.

Negative Numbers

Working with negative numbers can be tricky, but understanding some key principles makes it easier.

Firstly, remember that subtracting a positive number is the same as adding a negative number. For instance, \-8 - 64\ is the same as \-8 + (-64) = -72\.

Secondly, when you multiply two negative numbers, the result is positive. When you multiply a negative number by a positive one, the result is negative. In our example, multiplying 3 by -2 gives -6.

Lastly, dividing two negative numbers yields a positive result. For our example, \(\frac{-72}{-6} = 12\). The rule is:

• Negative ÷ Negative = Positive
  
• Positive ÷ Negative or Negative ÷ Positive = Negative
  

Understanding these rules about negative numbers simplifies many algebraic problems and helps in finding correct solutions.