Problem 101 - Elementary Algebra

Question

For exercises 101-104, evaluate. Use a scientific or a graphing calculator. $$ -80 \div(-2)+4(-6)^{2} $$

Step-by-Step Solution

Answer

184

Step 1: Simplify Division

First, solve the division inside the expression: \[ -80 \, \div \, (-2) \] This equals to 40 because dividing a negative number by another negative number gives a positive result.

Step 2: Evaluate the Power

Next, value the exponent in the expression: \[ (-6)^2 \] This equals 36 because squaring a negative number results in a positive number.

Step 3: Multiply by the Coefficient

Now, multiply the result from step 2 by 4: \[ 4 \, \times \, 36 \] This equals 144.

Step 4: Add the Results

Finally, add the results from step 1 and step 3: \[ 40 \, + \, 144 \] This equals 184.

Key Concepts

Division of Negative Numbers

When you divide negative numbers, it's important to remember the rule: the division of two negative numbers results in a positive number. Think of negative signs as indicating the direction. When both numbers go in the same direction, they cancel each other out, resulting in a positive value.

In our example, we have \( -80 \div -2 \). Here’s step-by-step what happens:
- First, ignore the negative signs and divide 80 by 2, which gives you 40.
- Since both original numbers were negative, and two negatives make a positive, the answer is +40. So, \( -80 \div -2 = 40 \).

Exponents

Exponents indicate how many times a number, the base, is multiplied by itself. When you see an expression like \( (-6)^2 \), you're squaring the number -6. Here's how it works:

- Multiplication: \( (-6) \times (-6) \).
- Remember, a negative times a negative results in a positive.
- Therefore, \( (-6)^2 = 36 \).

Why is it positive? Because a negative number multiplied by itself an even number of times (like squaring) will always result in a positive number.

Multiplication and Addition

In algebra, the order of operations is crucial. Using the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division (left to right), Addition and Subtraction (left to right)), we follow a specific order to solve expressions.

After evaluating the exponent \( (-6)^2 \) to get 36, we multiply:
\( 4 \times 36 = 144 \). This step is about applying the multiplication before we proceed to addition.

Finally, add the results from previous steps:
From our division step: 40
From our multiplication step: 144
Add these together: \( 40 + 144 = 184 \).

Always keep in mind to follow the right order to avoid errors in your calculations. Simple mistakes can often be traced back to skipping steps or performing them out of the intended sequence.