Problem 42 - Prealgebra

Question

The problems below review the material on exponents we have covered previously. Expand and simplify. $$(-2)^{3}$$

Step-by-Step Solution

Answer

The expression \((-2)^3\) simplifies to \(-8\).

Step 1: Understand the Problem Statement

The expression \((-2)^3\) involves raising \(-2\) to the power of 3. This means we need to multiply \(-2\) by itself a total of 3 times.

Step 2: Multiply the Base

To expand \((-2)^3\), we calculate \((-2) \times (-2) \times (-2)\). Begin by multiplying the first two \(-2\)s: \((-2) \times (-2) = 4\).

Step 3: Multiply the Result by the Remaining Base

Now, take the result from Step 2 and multiply it by the remaining \(-2\): \(4 \times (-2) = -8\).

Step 4: Conclusion

The expression \((-2)^3\) simplifies to \(-8\).

Key Concepts

Negative Bases

When dealing with exponents, it's common to see numbers raised to a power, and things get even more interesting when the base is a negative number. Let's break down what happens when we raise a negative number to an exponent. In the expression \((-2)^3\), \(-2\) is our base, and 3 is the exponent. Essentially, \((-2)^3\) tells us to multiply \(-2\) by itself three times. Each multiplication will affect the sign of the product depending on whether the exponent is even or odd.

• If the exponent is even, the negative sign effectively "cancels out," leaving us with a positive result. For instance, \((-2)^2 = (-2) \times (-2) = 4\).
• If the exponent is odd, as in this exercise, the result will retain the negative sign. Therefore, \((-2)^3 = (-2) \times (-2) \times (-2) = -8\).

So, raising negative numbers to odd powers results in a negative product, maintaining that original negative sign.

Multiplication of Integers

Understanding the multiplication of integers is crucial, especially when negative numbers are involved. Integers can be positive, negative, or zero, and multiplying them follows specific rules that determine the sign of the outcome.

Rules for Multiplying Integers

• **Positive Times Positive**: The product is positive. For example, \(2 \times 3 = 6\).
• **Negative Times Negative**: Two negatives make a positive. Example: \((-2) \times (-3) = 6\).
• **Positive Times Negative** or **Negative Times Positive**: The result is negative. Example: \(2 \times (-3) = -6\) or \((-2) \times 3 = -6\).

In the case of \((-2)^3\), the multiplication sequence \((-2) \times (-2) = 4\) (since two negatives make a positive), and then multiplying the positive 4 by the next \(-2\) returns it to a negative \(-8\).
Being familiar with these basic rules makes it much easier to handle expressions involving multiple negative integers.

Simplifying Expressions

Simplifying expressions involves performing operations in an orderly manner to condense an expression to its simplest form. With exponents, simplifying often means expanding and combining identical factors.

Steps in Simplifying Exponential Expressions

• **Expand the Expression**: Write it out as multiplication (e.g., \((-2)^3 = (-2) \times (-2) \times (-2)\)).
• **Perform Multiplications**: Follow the order of operations, often starting by multiplying pairs.
• **Combine Results**: Once multiplied, assess the product's sign and value.

For \((-2)^3\), after expanding, you multiply step by step, recognizing how each multiplication affects the sign of the outcome. The goal is to simplify the expression to a single integer: in this case, \(-8\).
Simplifying like this makes complex expressions more approachable and calculable.