Problem 67 - Algebra A Combined Function

Question

Perform the indicated operation. \(-1^{7}\)

Step-by-Step Solution

Answer

The result is -1.

Step 1: Understand the Expression

The expression \(-1^7\) is referring to the number \(-1\) raised to the power of \(7\), which means we need to multiply \(-1\) by itself 7 times.

Step 2: Multiply -1 by Itself Odd Number of Times

Since \(7\) is an odd number, multiplying \(-1\) by itself will alternate between \(-1\) and \(1\) with each multiplication. Odd powers of \(-1\) always result in \(-1\).

Step 3: Calculate

Let's perform the multiplication: \(-1\times-1=1\), \(1\times-1=-1\), \(-1\times-1=1\), \(1\times-1=-1\), \(-1\times-1=1\), and finally \(1\times-1=-1\). Thus, the result of \(-1^7\) is \(-1\).

Key Concepts

Understanding Negative Numbers

Negative numbers are numbers less than zero, often represented with a minus sign (−). They are crucial in mathematics because they allow us to extend the number line in the opposite direction. For example, if you picture a graph or a ruler, negative numbers are positioned left of zero.
Negative numbers have interesting properties, especially when it comes to operations like addition, multiplication, and exponents. A unique feature of negative numbers is how their sign changes based on whether they are multiplied by an odd or even number of negatives.

• If you multiply two negative numbers together, the result is positive.
• When you multiply a negative number with a positive number, the result is still negative.
• If there’s an odd number of negative numbers, the final product is negative.

In our exercise, \(-1^7\), we are dealing with multiplication across the odd power 7, which is why it's essential to focus on the properties of negative numbers.

Odd Powers and Their Effect on Negatives

When a number, like \(-1\), is raised to a power, it means multiplying \(-1\) by itself several times. In mathematical terms, these are called ‘exponents,’ with the number of times you multiply it by itself being the ‘power’. Odd powers are any power that is an odd number, such as 1, 3, 5, 7, etc.
The effect of raising negative numbers to odd powers is distinct. Each round of multiplication can alter the sign of the result. Let's break it down with our example of \(-1^7\):

• The first multiplication: \(-1 \times -1 = 1\)
• Second: \(1 \times -1 = -1\)

This pattern repeats, flipping the sign every time another \(-1\) is multiplied.

• Since \(7\) is odd, seven flips leaves us with a negative outcome.

Thus, any negative number raised to an odd power will remain negative.

Multiplication Involving Negative Numbers

Multiplication is a core operation that combines repeated addition. When multiplying negative numbers, we not only consider the numerical result but also the sign of the result. As mentioned, two negatives result in a positive product, and a negative and a positive results in a negative product.
Our example, \(-1^7\), involves multiplying \(-1\) seven times:

• Step 1: \(-1 \times -1 = 1\),
• Step 2: \(1 \times -1 = -1\), alternating because each multiplication by \(-1\) changes the sign.
• Following this pattern, you'll see the sign flip \(7\) times, ending on a negative result.

By understanding these patterns in multiplication, particularly with negative numbers, we can solve complex problems and make predictions about outcomes for different powers and arrays of numbers.