Problem 48 - Algebra for College Students

Question

Simplify each of the numerical expressions. $$(-2)^{3}+2(-2)^{2}-3(-2)-1$$

Step-by-Step Solution

Answer

The simplified expression is 5.

Step 1: Simplify the Exponents

Evaluate the exponent expressions. For \((-2)^3\):\[(-2) imes (-2) imes (-2) = -8\]For \[(-2)^2\]:\[(-2) imes (-2) = 4\]Then substitute these values into the expression:\[-8 + 2(4) - 3(-2) - 1\]

Step 2: Simplify Multiplications

Next, simplify any remaining multiplication within the expression. Multiply the values:\[2(4) = 8\]\[-3(-2) = 6\]Substituting these, the expression becomes:\[-8 + 8 + 6 - 1\]

Step 3: Combine the Terms

Now, add and subtract the numbers in the expression:First, \(-8 + 8 = 0\)Then, \(0 + 6 = 6\)Finally, \(6 - 1 = 5\)Thus, the simplified expression is 5.

Key Concepts

Exponents

Exponents are a way to express repeated multiplication of a number by itself, often referred to as the "power" of a number. In the given expression, we have two instances of exponents, namely

• \((-2)^3\): This means that the base, -2, is multiplied by itself three times resulting in: \((-2) \times (-2) \times (-2) = -8\).
• \((-2)^2\): This denotes that the base, -2, is multiplied by itself twice, yielding: \((-2) \times (-2) = 4\).

Using exponents simplifies complex expressions by condensing repeated multiplication into a more manageable notation. Remember, **raising a negative number to an odd power results in a negative number**, while **raising a negative number to an even power results in a positive number**. It is important to keep these rules in mind to evaluate expressions accurately.

Multiplication

In simplifying expressions, the next step after evaluating exponents is handling multiplication. The expression in question has terms that involve multiplication with numbers calculated from exponents:

• First, move on to \[2(4) = 8\]. This represents multiplying 2 by the result from \((-2)^2\), which we found to be 4.
• Next, simplify \[-3(-2) = 6\]. It involves multiplying -3 by -2, obtained when handling the term \(-3\times(-2)\).

Notice that negative multiplied by negative results in a positive. Multiplication helps combine terms and break down complex expressions into simpler parts, crucial for getting to the result.

Combining Like Terms

Once your expression no longer has any unresolved exponents or multiplication, the next step is to combine like terms. Like terms are numbers or expressions that contain the same variables raised to the same power. In our expression, we focus on combining the constant numbers we are left with:

• Start with \(-8 + 8\): This results in \(0\), as adding numbers with the same value but opposite signs cancels them out.
• Then move to \(0 + 6\): This simply equals \(6\).
• Finally, we solve \(6 - 1\) which leaves us with \(5\).

Combining like terms assists in reducing the expression to its most simple form by methodically adding or subtracting the constants.

Numerical Expressions

Numerical expressions involve constants and operations without unknown variables, meaning everything can be resolved using standard arithmetic operations. Understanding how to approach these expressions helps in systematically simplifying them. Our original expression:\[(-2)^3 + 2(-2)^2 - 3(-2) - 1\]required us to follow clear steps:

• Simplify exponents first to eliminate any repeated multiplications.
• Resolve multiplications next to combine factors and simplify terms where possible.
• Finally, combine resulting terms to arrive at a simplified numerical result.

Numerical expressions allow us to practice arithmetic manipulation, leading to understanding complex problems through a step-by-step process. Following these strategies ensures clarity and accuracy when solving such expressions.