Problem 10 - Introductory and Intermediate Algebra for College Students 4th

Question

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

Step-by-Step Solution

Answer

The solution for the exponential expression \((-1)^{6}\) is 1

Step 1: Identify the Expression

We need to evaluate or simplify: \((-1)^{6}\).

Step 2: Apply Order of Operations

Following the order of operations (PEMDAS/BODMAS), we simplify the expression step by step, handling parentheses, exponents, multiplication/division, then addition/subtraction.

Step 3: Compute the Result

Carrying out the operations systematically, we arrive at the result.

Step 4: State the Result

The solution for the exponential expression \((-1)^{6}\) is 1

Key Concepts

Understanding the Rules of Exponents

Exponents, also known as powers, are a fundamental part of algebra and mathematics. They allow us to express repeated multiplication of the same number in a simplified manner. When we talk about the rules of exponents, we are referring to the guidelines that dictate how we perform operations involving powers.

Here are some key rules:

• When multiplying like bases, you add the exponents: \(a^m \cdot a^n = a^{m+n}\).
• To divide like bases, you subtract the exponents: \(\frac{a^m}{a^n} = a^{m-n}\).
• Raising a power to another power involves multiplying the exponents: \( (a^m)^n = a^{m \cdot n}\).

These rules are vital for simplifying expressions and performing calculations efficiently. For example, if you have \((-1)^6\), it simplifies directly due to the exponent rules, indicating how a negative raised to an even power results in a positive outcome.

The Role of Even Exponents

Even exponents play a crucial role when it comes to determining the sign of an exponential expression. Whenever you raise a negative number to an even power, the result will be a positive number. This happens because multiplying two negative numbers always gives a positive product.

Let's explain further with an example:

• Consider \((-1)^2\), which equals \(1\) since \(-1 \times -1 = 1\).
• When increasing the power, \((-1)^4 = (-1)^2 \times (-1)^2 = 1 \).

Thus, \((-1)^6\) equals \(1\), since each pair of \(-1\ imes -1\) keeps the result positive. This characteristic is consistent across all even exponents.

Calculating Powers in Simple Steps

Calculating powers involves taking a base number and multiplying it by itself a certain number of times, as indicated by the exponent. Let's walk through the calculation process using \((-1)^6\) as an example.

• First, identify the base and the exponent: here, \(a = -1\) and \(n = 6\).
• Recognize that \((-1)^6\) means multiplying \(-1\) by itself six times: \(-1 \times -1 \times -1 \times -1 \times -1 \times -1\).
• Pair them up: every two \(-1\)’s give you \(1\).
• Since \(6\) is even, the final product is \(1\).

The computation underscores the importance of understanding how powers work, particularly when dealing with negative bases and even exponents, ensuring you get the right sign.