Problem 34 - Algebra for College Students

Question

Simplify each of the numerical expressions. $$4(-2)^{3}-3(-1)^{4}$$

Step-by-Step Solution

Answer

The simplified expression is -35.

Step 1: Simplify the Exponentiation

First, simplify the expressions that involve exponents. Begin with \((-2)^{3}\) which means \((-2)\times(-2)\times(-2)\). Calculate as follows:- \(-2)\times(-2) = 4\).- \(4\times(-2) = -8\). Therefore, \((-2)^{3} = -8\). Next, consider \((-1)^{4}\) which means \((-1)\times(-1)\times(-1)\times(-1)\). Calculate as follows:- \((-1)\times(-1) = 1\).- \(1\times(-1) = -1\).- \(-1\times(-1) = 1\). Therefore, \((-1)^{4}=1\).

Step 2: Apply Multiplication

Now, substitute the simplified values back into the expression and multiply: For \(4(-2)^{3}\): Substitute \((-2)^{3}\) with \(-8\), yielding \(4\times(-8)\), which gives \(-32\).For \(-3(-1)^{4}\): Substitute \((-1)^{4}\) with \(1\), yielding \(-3\times 1\), which is \(-3\).

Step 3: Perform Addition/Subtraction

Finally, perform the addition/subtraction to simplify the expression fully.Take the two products from the previous step and combine them: \(-32 - 3 = -35\). This gives the final simplified expression.

Key Concepts

Exponentiation

Exponentiation is a mathematical operation involving numbers raised to a power. When we see something like \((-2)^3\), it means we take \(-2\) and multiply it by itself a total of three times.
It's crucial to pay attention to the negative sign here, since

• If an exponent is odd, the result retains the sign of the base. In this context, \((-2)^3 = (-2) \times (-2) \times (-2) = -8\).
• When an exponent is even, any number becomes positive. For example, when calculating \((-1)^4\), it results in \(1\) because \((-1) \times (-1) \times (-1) \times (-1) = 1\). The negative signs cancel out in pairs, making the result positive.

Understanding this process helps clarify how signs and bases interact in exponentiation. Recognizing this, when simplifying expressions, tackle this step first as it sets the stage for subsequent operations.

Multiplication and Division

After exponentiation, the next task is handling multiplication (or division if it appears). These operations take precedence over addition and subtraction due to the order of operations.
Using our simplified exponentiation results from before, replace the exponential terms in the expression. We have:

• For \(4(-2)^3\), substitute \(-8\) for \((-2)^3\), resulting in \(4 \times (-8)\), which simplifies to \(-32\).
• For \(-3(-1)^4\), substitute \(1\) for \((-1)^4\), leading to \(-3 \times 1\), which simplifies to \(-3\).

The product of numbers requires simply multiplying the integers, being mindful of retaining correct sign based on multiplication rules, where a positive multiplied by a negative results in a negative.

Addition and Subtraction

Once multiplication and division are sorted out, only addition and subtraction remain. These should be performed last according to the order of operations (or PEMDAS/BODMAS rules, which stand for Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).
Now, focus on adding and subtracting the results obtained from earlier multiplication. Our expression now is

• \(-32\) and \(-3\).

Since both numbers are negative, we can think of stacking them more negatively. Therefore, compute \(-32 - 3\), which proceeds as simply adding \(3\) more to \(-32\), resulting in \(-35\).
The result is straightforward, but it highlights the importance of understanding how to manage addition and subtraction of negatives. Thus, properly simplifying numerical expressions ensures all components align correctly.