Problem 36 - Elementary Algebra

Question

\((-3)^{3}\)

Step-by-Step Solution

Answer

-27

Step 1: Understand the Problem

The given problem is \((-3)^3\). This means raising -3 to the power of 3.

Step 2: Apply the Exponent

Raising a number to the power of 3 means multiplying that number by itself three times: \((-3) \times (-3) \times (-3)\).

Step 3: Multiply Step-by-Step

First, multiply the first two numbers: \(-3 \times -3 = 9\). Next, multiply the result by the remaining number: \(9 \times -3 = -27\).

Step 4: Interpret the Result

After performing the multiplication, the final result of \((-3)^3\) is \(-27\).

Key Concepts

Negative Numbers

Negative numbers are numbers less than zero. They are usually represented with a minus sign (-). Understanding how they behave in different mathematical operations is crucial for solving equations correctly.

In the context of exponentiation, a negative base can significantly affect the outcome. For example, when a negative number is raised to an odd power, the result is negative. For even powers, the result is positive. This is because:

• Odd powers: The signs do not cancel out (e.g., \textcolor{red}{embracing} \textcolor{red}{\((-3)^3\) gives -27}.
• Even powers: The signs cancel out (e.g., \textcolor{red}{\((-3)^2\)} equals 9).

Handling negative numbers in algebra is much easier when you remember these rules. When multiplying or dividing them, multiply the absolute values and assign the sign based on whether you're working with an even or odd number of negatives.

Multiplication

Multiplication is one of the basic arithmetic operations. It combines multiple groups of a number into one total amount. For instance, 2 multiplied by 3 means adding 2 three times: 2 + 2 + 2 = 6.

When multiplying with negative numbers, it's important to keep track of the signs:

• Negative times positive: \textcolor{red}{The result is negative} (e.g., \textcolor{red}{\(-3 \times 3\)} = -9).
• Negative times negative: \textcolor{red}{The result is positive} (e.g., \textcolor{red}{\(-3 \times -3\)} = 9).

Let's apply this to our example. First, we multiply the first two numbers in the product: \textcolor{red}{\((-3) \times (-3)\)} gives 9. Then, take that result and multiply it with the remaining number: \textcolor{red}{\(9 \times -3\)} = -27.

Remember that keeping track of signs is essential in multiplication, especially when dealing with negative numbers.

Power of 3

Exponentiation is a mathematical operation where a number (the base) is multiplied by itself a specified number of times (the exponent). In this exercise, the exponent is 3.

A power of 3 means you multiply the base number by itself three times. So, \textcolor{red}{\((-3)^3\)} can be broken down as follows: \textcolor{red}{\((-3) \times (-3) \times (-3)\)}. Here’s a step-by-step guide:

1. Multiply the first two numbers: \textcolor{red}{\((-3) \times (-3)\)} = 9 (since negative times negative is positive).
2. Multiply the result by the third number: \textcolor{red}{\(9 \times (-3)\)} = -27 (since positive times negative is negative).

So, \textcolor{red}{\((-3)^3\)} = -27.