Problem 26 - Algebra A Combined Function

Question

Evaluate each expression. $$ (-5)^{3} $$

Step-by-Step Solution

Answer

The result of \((-5)^{3}\) is -125.

Step 1: Understanding the Expression

Let's start by understanding the expression we need to evaluate. The given expression is \((-5)^{3}\), which means we need to multiply \(-5\) by itself a total of three times.

Step 2: Calculate the First Multiplication

First, multiply \(-5\) by \(-5\). This equals \((-5) \times (-5) = 25\) because multiplying two negative numbers results in a positive number.

Step 3: Calculate the Second Multiplication

Now, multiply the result from the first multiplication (25) by \(-5\) to compute the cube: \(25 \times (-5) = -125\). Multiplying a positive number by a negative number results in a negative number.

Key Concepts

Negative numbers

Negative numbers are numbers less than zero. They are typically written with a minus sign in front, like \(-5\). These numbers can be tricky, especially when working with exponents. But, understanding them is essential as they are used often in everyday calculations.
When multiplying negative numbers:

• If you multiply two negative numbers together, the result is positive. For example, \( (-5) \times (-5) = 25 \).


• If you multiply a negative number by a positive number, the result is negative, such as \( 25 \times (-5) = -125 \).

Working with negatives becomes easier once you grasp these multiplication rules. Every negative number behaves the same way, so these patterns are consistent.

Multiplication

Multiplication is a mathematical operation where a number is added to itself a specific number of times. For instance, \((-5)^{3}\) means we multiply \(-5\) by itself three times: \((-5) \times (-5) \times (-5)\). This is where the magic happens with exponents as well!
The first step in solving \((-5)^{3}\) involves multiplying \(-5\) by \(-5\). When you do this, \( (-5) \times (-5) = 25 \), turning it positive.
Next, you take this new value, \(25\), and multiply it by \(-5\) again: \((25) \times (-5) = -125\). Here, keeping track of negative signs is important.
Understanding multiplication, especially with negatives, involves paying attention to these positive and negative changes with each step.

Cube of a number

The cube of a number means multiplying the number by itself, and then by itself once more. It involves raising a number to the power of three. For \(-5\), cubing it is written as \( (-5)^{3} \), which means \(-5 \times -5 \times -5\).
In practical terms:

• Start with \(-5\).
• First, multiply \(-5 \times -5 = 25\). This step removes the negative because multiplying two negatives results in a positive.


• Then, multiply the result, \(25\), by \(-5\), giving \(-125\). Here, you change back to a negative.

Cubing a negative number will always lead to a negative result when repeated, odd-numbered multiplications. It's a fascinating example of how exponents can expand our understanding of multiplication.