Problem 9 - Algebra 1: Concepts and Skills

Question

Find the product. \((-1)^{5}\)

Step-by-Step Solution

Answer

-1

Step 1: Understand the Base and the Exponent

In this expression, \((-1)^{5}\), -1 is the base and 5 is the exponent. We can see that 5 is an odd number.

Step 2: Apply the Exponentiation Rule for Negative Base

A negative base raised to an odd exponent will always yield a negative result. Here, as the base is -1 and the exponent is odd (5), the result will be -1.

Step 3: Write Down the Final Result

So, the product of \((-1)^{5}\) is -1.

Key Concepts

Negative Numbers

Negative numbers are an integral part of mathematics, representing values less than zero. They are marked with a minus sign (-) and have unique properties when involved in calculations, especially in exponentiation.

• When a negative base is raised to a power, its result depends on whether the exponent is odd or even. This is because multiplying negative numbers can switch the sign based on pairings.
• For example, \[(-1)^2 = (-1) imes (-1) = 1\]. Here, two negatives make a positive, since the exponent (2) is even.
• Conversely, \[(-1)^3 = (-1) imes (-1) imes (-1) = -1\]. Three negatives leave you with a negative result, as the exponent (3) is odd.

Understanding negative numbers helps lay the groundwork for their interplay with odd and even numbers in exponentiation.

Odd Exponents

Odd exponents are whole numbers like 1, 3, 5, etc., which fundamentally affect the outcome when used in power expressions with negative bases.

• An odd exponent results in a negative product if the base is negative. This happens because there remains an odd number of negative factors, leading to a negative sign.
• For example, consider \((-2)^5\). Expanding this, you have five -2s multiplied together:\[(-2) imes (-2) imes (-2) imes (-2) imes (-2) = -32\], still negative because the count of -2 multiplications is odd.
• This property is specific to negative bases and is pivotal in algebra and beyond, ensuring accurate computations.

Odd exponents are key for maintaining the sign when working with negative bases.

Exponential Rules

Exponential rules simplify expressions involving exponents by providing consistent principles for calculations. These govern how to handle powers in diverse scenarios.

• One key rule is when a negative number is raised to an exponent: the sign of the result aligns with whether the exponent is odd or even.
• For instance, the expression \[(-1)^5 = -1\] follows these rules, since 5 is odd and the base -1 remains negative.
• Another important rule is the product of powers: \(a^m \times a^n = a^{m+n}\). However, this usually applies to positive bases unless expanded carefully with negatives.

Understanding and applying exponential rules streamline complex mathematical operations, making it easier to predict results and solve problems efficiently.