Problem 12 - Introductory and Intermediate Algebra for College Students 4th

Question

Evaluate each exponential expression. $$-1^{6}$$

Step-by-Step Solution

Answer

\(-1^{6} = 1\)

Step 1: Understand the expression

The expression given is \(-1^{6}\). Here, -1 is the base and 6 is the exponent.

Step 2: Apply Rules of Exponents

In an exponential expression \(a^n\), the base \(a\) is multiplied by itself \(n\) times. When \(n\) is an even number, and \(a\) is -1, the result will be 1 since \(-1 \times -1 = 1\). Thus \(-1^{6}\) is equal to \(1^{3}\) (since -1 multiplies by itself 6 times).

Step 3: Calculate the power

Now, we just need to calculate \(1^{3}\), which is equal to 1 (because any number to the power of 3 is that number multiplied by itself three times). Therefore, \(-1^{6} = 1^{3} = 1\).

Key Concepts

Base and Exponent

In mathematics, an exponent is a shorthand notation that represents repeated multiplication of the same number. In the expression \(-1^6\), -1 is called the base, and 6 is called the exponent. The exponent tells us how many times the base is multiplied by itself.

For example:

• If the base is 2 and the exponent is 3, \(2^3 = 2 \times 2 \times 2\).
• Similarly, if the base is -1 and the exponent is 6, it means we multiply -1 by itself six times.

The concept of base and exponent is crucial because it helps in simplifying complex calculations. Instead of writing \(-1 \times -1 \times -1 \times -1 \times -1 \times -1\), we can write \(-1^6\). Understanding this shorthand can make working with large and complex numbers more manageable.

Rules of Exponents

The rules of exponents are guidelines that help us perform operations involving powers efficiently. They make it easier to simplify expressions and solve exponent-related problems. Let’s take a look at some basic rules using the example of \(-1^6\):

• Product of Powers Rule: If you are multiplying two powers with the same base, you can add the exponents. For example, \(a^m \times a^n = a^{m+n}\).
• Power of a Power Rule: When raising a power to another power, multiply the exponents: \((a^m)^n = a^{m\times n}\).
• Power of a Product Rule: Distribute the exponent to each factor in the product: \((ab)^n = a^n \times b^n\).

In the case of \(-1^6\), the rule to remember is that if the exponent is even, the result of raising -1 to that power will always be 1. That's because pairs of -1's multiply to get one. Thus, \(-1^6 = (-1 \times -1) \times (-1 \times -1) \times (-1 \times -1) = 1 \times 1 \times 1 = 1\). Each pair makes a positive 1, leading to the overall result of 1.

Exponential Expressions

Exponential expressions involve numbers raised to a power. They are prevalent in algebra and are used extensively in fields such as physics, economics, and computer science. Understanding how to evaluate and simplify these expressions is key to solving many mathematical problems.

Consider an expression like \(-1^6\), which combines a negative base with an exponent. This is a type of exponential expression. Evaluating it involves applying the rules of exponents systematically.

• First, understand the base and exponent relationship (e.g., -1 raised to the 6).
• Apply relevant rules of exponents (e.g., any number raised to an even exponent will have a positive result if the base is negative).
• Simplify step-by-step to reach the final result.

Lastly, remember that knowing how to break down and understand each part of an exponential expression is vital. Practice and familiarity will help you to tackle increasingly wider ranges of exponential equations with confidence.