Problem 97 - Elementary Algebra

Question

For exercises 15-100, evaluate. $$ -2(6-9)^{2}-5 $$

Step-by-Step Solution

Answer

-23

Step 1 - Simplify Inside the Parentheses

Evaluate the expression inside the parentheses first. In this case, calculate the difference: \(6 - 9 = -3\)

Step 2 - Square the Result

Next, square the result of step 1: \((-3)^{2} = 9\)

Step 3 - Multiply by -2

Now, multiply the squared value by -2: \(-2 \times 9 = -18\)

Step 4 - Subtract 5

Finally, subtract 5 from the result of step 3: \(-18 - 5 = -23\)

Key Concepts

Order of Operations

When evaluating mathematical expressions, it's crucial to follow the order of operations. This helps ensure that everyone calculates the same value. The order of operations can be remembered by the acronym PEMDAS, which stands for:
- Parentheses
- Exponents (such as squaring numbers)
- Multiplication and Division (from left to right)
- Addition and Subtraction (from left to right).
For example, in the expression -2(6-9)^2 - 5, we work on what's inside the parentheses first, then move on to exponents, followed by multiplication, and finally perform subtraction.

Parentheses

Parentheses are used to group parts of an expression that should be evaluated first. In the given expression -2(6-9)^2 - 5, we first simplify the value inside the parentheses.
- Calculate 6 - 9 inside the parentheses to get -3.
It's important to always handle innermost parentheses first if there are multiple sets. This will guide you to the correct answer by ensuring calculations are done in the proper order.

Squaring Numbers

Squaring a number means multiplying it by itself. Once we have simplified inside the parentheses in the given expression -2(6-9)^2 - 5, we are left with -3.
- Squaring -3 means calculating (-3)^2 = 9.
Remember, squaring a negative number always results in a positive number. Knowing this prevents mistakes when dealing with negative bases.

Multiplication

After squaring the result, the next step is to multiply. For the expression -2(6-9)^2 - 5, after squaring, we get 9.
- Multiply -2 by 9 to get -18.
Remember to watch the signs when multiplying; a negative times a positive gives a negative product.

Addition and Subtraction

Finally, address any addition or subtraction. In our expression -2(6-9)^2 -5, after the multiplication step, we have -18.
- Subtract 5 from -18 to get -23.
This last step simplifies the expression completely. Always follow through these steps in order and ensure each is done correctly before moving to the next.