Identifying Positive And Negative ValuesDetermine Whether The Value Of Each Power Is Positive Or Negative.1. $(-10)^5$2. $(-3)^4$3. $(-2)^9$4. $(-1)^2$5. $(-6)^3$6. ( − 4 ) 8 (-4)^8 ( − 4 ) 8

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Introduction

In mathematics, exponents are a fundamental concept that helps us simplify complex expressions and solve equations. When dealing with negative numbers raised to a power, it's essential to understand the rules governing the sign of the result. In this article, we'll delve into the world of negative exponents and explore how to determine whether the value of each power is positive or negative.

The Rules of Negative Exponents

Before we dive into the examples, let's review the basic rules of negative exponents:

  • When a negative number is raised to an even power, the result is always positive.
  • When a negative number is raised to an odd power, the result is always negative.
  • When a negative number is raised to the power of 0, the result is always 1.

Examples and Solutions

1. (−10)5(-10)^5

To determine the sign of the result, we need to consider the exponent. In this case, the exponent is 5, which is an odd number. According to the rules, when a negative number is raised to an odd power, the result is always negative.

$(-10)^5 = -10 \times -10 \times -10 \times -10 \times -10 = -100000$

2. (−3)4(-3)^4

The exponent in this case is 4, which is an even number. According to the rules, when a negative number is raised to an even power, the result is always positive.

$(-3)^4 = 3 \times 3 \times 3 \times 3 = 81$

3. (−2)9(-2)^9

The exponent in this case is 9, which is an odd number. According to the rules, when a negative number is raised to an odd power, the result is always negative.

$(-2)^9 = -2 \times -2 \times -2 \times -2 \times -2 \times -2 \times -2 \times -2 \times -2 = -512$

4. (−1)2(-1)^2

The exponent in this case is 2, which is an even number. According to the rules, when a negative number is raised to an even power, the result is always positive. However, when the base is -1, the result is always 1, regardless of the exponent.

$(-1)^2 = 1$

5. (−6)3(-6)^3

The exponent in this case is 3, which is an odd number. According to the rules, when a negative number is raised to an odd power, the result is always negative.

$(-6)^3 = -6 \times -6 \times -6 = -216$

6. (−4)8(-4)^8

The exponent in this case is 8, which is an even number. According to the rules, when a negative number is raised to an even power, the result is always positive.

$(-4)^8 = 4 \times 4 \times 4 \times 4 \times 4 \times 4 \times 4 \times 4 = 65536$

Conclusion

In conclusion, understanding the sign of exponents is crucial when dealing with negative numbers raised to a power. By applying the rules of negative exponents, we can determine whether the value of each power is positive or negative. Remember, when a negative number is raised to an even power, the result is always positive, and when a negative number is raised to an odd power, the result is always negative. With practice and patience, you'll become proficient in determining the sign of exponents and solving complex equations with ease.

Additional Tips and Resources

  • To reinforce your understanding of negative exponents, try solving more examples and exercises.
  • Practice, practice, practice! The more you practice, the more comfortable you'll become with negative exponents.
  • If you're struggling with a particular concept or topic, don't hesitate to seek help from a teacher, tutor, or online resource.
  • For additional resources and practice problems, check out the following websites:

Q: What is the rule for negative exponents?

A: The rule for negative exponents states that when a negative number is raised to an even power, the result is always positive, and when a negative number is raised to an odd power, the result is always negative.

Q: How do I determine the sign of a negative exponent?

A: To determine the sign of a negative exponent, you need to consider the exponent itself. If the exponent is even, the result is positive, and if the exponent is odd, the result is negative.

Q: What happens when a negative number is raised to the power of 0?

A: When a negative number is raised to the power of 0, the result is always 1, regardless of the base.

Q: Can you give me an example of a negative exponent?

A: Yes, here's an example: (−3)4(-3)^4. In this case, the exponent is 4, which is an even number. Therefore, the result is positive: (−3)4=81(-3)^4 = 81.

Q: What about a negative exponent with an odd power?

A: Here's an example: (−2)9(-2)^9. In this case, the exponent is 9, which is an odd number. Therefore, the result is negative: (−2)9=−512(-2)^9 = -512.

Q: How do I simplify a negative exponent?

A: To simplify a negative exponent, you can use the rule for negative exponents. For example, (−5)6(-5)^6 can be simplified as follows:

(−5)6=(−5)×(−5)×(−5)×(−5)×(−5)×(−5)=15625(-5)^6 = (-5) \times (-5) \times (-5) \times (-5) \times (-5) \times (-5) = 15625

Q: Can you give me some practice problems to try?

A: Yes, here are some practice problems to try:

  1. (−4)7(-4)^7
  2. (−2)3(-2)^3
  3. (−6)2(-6)^2
  4. (−1)5(-1)^5
  5. (−3)9(-3)^9

Q: What if I get stuck on a problem?

A: Don't worry! If you get stuck on a problem, try breaking it down into smaller steps. You can also ask a teacher, tutor, or classmate for help. Additionally, you can try using online resources, such as Khan Academy or Mathway, to get additional support.

Q: How can I apply negative exponents in real-life situations?

A: Negative exponents can be applied in a variety of real-life situations, such as:

  • Calculating the area of a rectangle with negative dimensions
  • Determining the volume of a cube with negative side length
  • Solving problems involving negative interest rates or investments

By understanding and applying negative exponents, you can solve a wide range of problems and become more confident in your math skills.

Conclusion

In conclusion, negative exponents can seem intimidating at first, but with practice and patience, you can master this concept and become proficient in solving complex equations. Remember to apply the rules for negative exponents, simplify expressions, and practice, practice, practice! With these tips and resources, you'll be well on your way to becoming a math whiz.