Problem 87 - Elementary and Intermediate Algebra

Question

Evaluate each expression. $$ -2(-1)^{2}+3(-1)-3 $$

Step-by-Step Solution

Answer

The expression evaluates to -8.

Step 1: Evaluate the Exponent

First, solve the expression inside the exponent. Since (-1)^2 means multiplying -1 by itself, we get (-1) imes (-1) = 1.

Step 2: Multiply with the Outside Coefficient

Now, take the result from the exponent step and multiply it by the coefficient in front. So, multiply -2 imes 1 = -2.

Step 3: Multiply the Variables

Next, calculate the multiplication of 3(-1). This results in 3 imes (-1) = -3.

Step 4: Combine All Parts

Now, combine all the parts: -2 + (-3) - 3. This simplifies to: -2 - 3 - 3.

Step 5: Simplify the Expression

Finally, add the numbers together from left to right. Starting with -2 - 3, we get -5. Then, -5 - 3 gives us the final answer of -8.

Key Concepts

Exponentiation

Exponentiation is a mathematical operation that involves raising a number, known as the base, to a certain power, represented as the exponent. For instance, in the provided exercise, the expression (-1)^2 is given. This tells us to multiply -1 by itself two times, because the exponent is 2.
\[(-1) imes (-1) = 1\]
Exponentiation can be easily tackled by remembering the basic rules:

• Negative base with an even exponent: Always results in a positive outcome, since multiplying two negatives makes a positive.
• Negative base with an odd exponent: Results in a negative outcome.

Understanding these rules helps simplify and solve such expressions accurately.

Coefficient Multiplication

Coefficient multiplication involves multiplying the result of the exponentiation with a number that acts as a multiplier—the coefficient in front of the expression. In this exercise, the coefficient is -2, which multiplies the result of the exponentiation:
\[ -2 imes 1 = -2 \]
Key points to consider when dealing with coefficient multiplication include:

• Sign of the Coefficient: A negative coefficient will invert the sign of the product relative to the multiplicand.
• Order of Operations: Ensure that the exponentiation is performed before multiplication—as dictated by mathematical operations' hierarchy (PEMDAS/BODMAS).

Remembering these steps assures that the expression is calculated accurately.

Combining Like Terms

Combining like terms is an essential step when you simplify algebraic expressions. It involves adding or subtracting coefficients of terms that have identical variables or no variables at all. In the expression -2 - 3 - 3, all terms are constants, meaning they have no variables. Therefore, they can be directly combined:
Starting with the leftmost term, we add -2 and -3 to get:
\[ -2 - 3 = -5 \] Then, add -5 and -3:
\[ -5 - 3 = -8 \]
Combining like terms reduces complexity and is crucial in deriving the simplest possible form of an expression.