Problem 78 - Intermediate Algebra

Question

Evaluate each expression. See Example \(9 .\) $$ (-2)^{3}\left(\frac{-6}{-2}\right)(-1) $$

Step-by-Step Solution

Answer

The evaluated expression equals 24.

Step 1: Simplify the First Expression

Begin by evaluating the expression \((-2)^{3}\). The operation here is raising \(-2\) to the power of 3, which means multiplying \(-2\) by itself three times: \((-2) \times (-2) \times (-2)\).Calculate:\((-2) \times (-2) = 4\)\(4 \times (-2) = -8\)So, \((-2)^{3} = -8\).

Step 2: Evaluate the Fraction

Evaluate the fraction expression, \(\frac{-6}{-2}\). Since both the numerator and the denominator have negative signs, the negatives cancel each other out, resulting in a positive value. Divide:\(-6 \div -2 = 3\).

Step 3: Evaluate the Third Expression

The third part of the expression is \((-1)\). This remains \(-1\), as it is a constant not subject to any further operation at this point.

Step 4: Combine All Three Results

Multiply the results from steps 1 to 3 together:1. Result from Step 1: \(-8\) 2. Result from Step 2: \(3\) 3. Result from Step 3: \(-1\)Now perform the multiplication: \((-8) \times 3 \times (-1)\)Calculate:\(-8 \times 3 = -24\)\(-24 \times (-1) = 24\)

Key Concepts

Exponentiation

In mathematics, exponentiation is a powerful operation. It involves raising a number, known as the base, to the power of another number, called the exponent. The exponent indicates how many times the base is multiplied by itself. Let's consider this with an example. Suppose we have the expression \((-2)^3\). This means we will multiply \(-2\) by itself three times:

• First, calculate \((-2) \times (-2) = 4\).
• Next, multiply that result by \(-2\) again: \(4 \times (-2) = -8\).

So, \((-2)^3 = -8\). Notice how the negative base multiplied an odd number of times results in a negative number. Keep this in mind: if the exponent is even, the result is positive; if it's odd, the result is negative.

Fraction Simplification

Simplifying fractions is a crucial part of solving math problems. It involves reducing the fraction to its simplest form. This often makes calculations easier and results cleaner. In our exercise, we have the fraction \(\frac{-6}{-2}\). Because both the numerator (the top number) and the denominator (the bottom number) are negative, those negatives effectively cancel each other out. Here's how it works:

• Negative divided by negative equals positive, so: \(-6 \div -2 = 3\).

Therefore, the simplified form of \(\frac{-6}{-2}\) is \(3\). Remember, the rules for dividing negative numbers apply here, ensuring the resulting fraction is positive.

Multiplying Negative Numbers

Understanding how to multiply negative numbers is essential for integer operations. When performing multiplication, the rules concerning negative numbers are straightforward:

• A negative times a negative is a positive.
• A positive times a negative is a negative.
• A negative times a positive is a negative.

Applying this to our exercise, first we have \((-8) \times 3\):

• Since \(-8 \times 3 = -24\), the result is negative because we're multiplying a negative by a positive.

Next, consider \(-24 \times (-1)\):

• Here, multiplying two negative numbers \(-24\) and \(-1\) gives us a positive result of \(24\).

Observe how the signs change depending on the number of negative factors you're dealing with. Once again, two negatives make a positive. Always remember these foundational rules to navigate integer operations effortlessly.