Problem 103 - Elementary and Intermediate Algebra

Question

Evaluate each expression. $$ (-3)^{3}\left(\frac{-4}{2}\right)(-1) $$

Step-by-Step Solution

Answer

The expression evaluates to -54.

Step 1: Evaluate the Power

First, calculate \((-3)^{3}\). This means multiplying \(-3\) by itself three times: \(-3 imes -3 imes -3 = 27 imes -3 = -27\).

Step 2: Simplify the Fraction

Simplify the fraction \(\frac{-4}{2}\). This is equivalent to \(-4 \div 2 = -2\).

Step 3: Combine the Components

Now multiply the results from the previous steps together. Start by multiplying \(-27\) from Step 1 by \(-2\) from Step 2: \(-27) \times (-2) = 54\).

Step 4: Multiply by the Remaining Factor

Finally, multiply the result from Step 3 by \(-1\): \(54 \times (-1) = -54\). This is the final result.

Key Concepts

Evaluating Expressions

Evaluating algebraic expressions is like solving a puzzle where each piece must be placed in the correct order. We need to simplify and solve the expression by working through each operation carefully. To solve the expression \((-3)^{3}\left(\frac{-4}{2}\right)(-1)\), we aim to simplify it step by step, according to the rules, so we reach a final numerical answer.

When evaluating expressions, it's essential to follow the order of operations, often abbreviated as PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). This guides us on which operations to perform first. In our exercise, our first task is to handle the exponents before multiplying or dividing the numbers.

Exponents

Exponents represent repeated multiplication. In our expression \((-3)^{3}\), the exponent \(3\) indicates that the base \(-3\) is multiplied by itself three times. This can be calculated as \((-3) \times (-3) \times (-3)\).

It's important to note that multiplying negative numbers follows its own rules:

• When two negative numbers are multiplied, the result is positive. For example, \((-3) \times (-3) = 9\).
• However, when you multiply three negative numbers, as we do here, the overall result is negative because \((-3) \times (-3) \times (-3) = -27\).

Understanding these rules helps simplify expressions involving exponents correctly and ensures accurate results.

Multiplication of Integers

Multiplying integers is straightforward but requires careful attention to the signs. It’s essential to know how to handle positive and negative numbers:

• Multiplying two positive numbers or two negative numbers results in a positive number. For example, \(2 \times 3 = 6\) and \((-4) \times (-2) = 8\).
• Multiplying a positive number by a negative number results in a negative number, such as \(2 \times (-3) = -6\).

In our expression \(-27 \times -2 \times -1\), step by step, we multiply \(-27\) and \(-2\) to get \(54\), and then multiply \(54\) by \(-1\) to get \(-54\).

Each multiplication step affects the sign and value of the accumulated result, which is why keeping track of signs is crucial when multiplying integers.