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Chapter 1: Problem 10

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

Short Answer

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

Step by step solution

01

Apply Rules of Exponents

Bearing in mind the rule that an even exponent results in a positive outcome, simply calculate the expression to determine its value. We raise -1 to the power of 6. Expressed in the form \(a^n\) where \(a\) is the base and \(n\) is the exponent, in this case \(a = -1\) and \(n = 6\). Calculate as follows: \[(-1)^6 = 1\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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 \(-1\)鈥檚 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.

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Most popular questions from this chapter

In Exercises \(109-116\), write a numerical expression for each phrase. Then simplify the numerical expression by performing the given operations. The difference between \(-11\) and the quotient of 20 and \(-5\)

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. I found the sum of \(-13\) and 4 by thinking of temperatures above and below zero: If it's 13 below zero and the temperature rises 4 degrees, the new temperature will be 9 below zero, so \(-13+4=-9\)

Explain how to simplify \(4 x^{2}+6 x^{2} .\) Why is the sum not equal to \(10 x^{4} ?\)

Explain how to multiply two real numbers. Provide examples with your explanation.

From here on, each exercise set will contain three review exercises. It is essential to review previously covered topics to improve your understanding of the topics and to help maintain your mastery of the material. If you are not certain how to solve a review exercise, turn to the section and the worked example given in parentheses at the end of each exercise. Consider the set $$\\{-6,-\pi, 0,0, \overline{7}, \sqrt{3}, \sqrt{4}\\}$$ List all numbers from the set that are a. natural numbers, b. whole numbers, c. integers, d. rational numbers, e. irrational numbers, f. real numbers. (Section 1.3 Example 5 )
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