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Chapter 5: Problem 124

In your own words, explain when \((-3)^{n}\) is positive and when it is negative.

Short Answer

Expert verified
(-3)^{n} is positive when n is even, and negative when n is odd.

Step by step solution

01

Determine the Sign Rule

To understand when (-3)^{n} is positive or negative, we need to determine the effect of n on the sign of the expression. Multiplying an odd number of negatives results in a negative product, whereas an even number of negatives leads to a positive product.
02

Check for Even and Odd Exponents

Since -3 is a negative number, (-3)^{n} will follow this rule: - If n is an even number, the product (-3)^{n} will be positive because multiplying an even count of negatives makes it positive ( (-3)^{2} = 9, (-3)^{4} = 81, etc. ). - If n is an odd number, the product (-3)^{n} will be negative because an odd count of negatives remains negative ( (-3)^{1} = -3, (-3)^{3} = -27, etc. ).

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Key Concepts

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

Negative Numbers
Negative numbers are numbers that are less than zero. They are typically represented with a minus sign (-) in front of them. When we talk about negative numbers, it's important to highlight that they are the opposite of positive numbers and have a multitude of applications in real life, like describing temperatures below zero or financial losses.
In mathematics, combining negative numbers follows specific rules:
  • Adding a negative number is the same as subtraction. For example, 5 + (-3) equals 2.
  • Subtracting a negative number is the same as addition. For example, 5 - (-3) equals 8.
  • Multiplying two negative numbers results in a positive number. For instance, (-2) x (-3) equals 6.
  • Dividing two negative numbers also results in a positive number, just like multiplying.
Understanding these rules is crucial, as they lay the groundwork for more complex operations involving exponents and algebraic equations.
Odd and Even Exponents
Exponents represent repeated multiplication of the same number. When raising numbers to an exponent, one must consider whether the exponent is odd or even, especially when dealing with negative bases like (-3)^{n} .
Here's a quick guide:
  • **Even Exponents:** When the exponent is even, such as 2, 4, 6, etc., raising a negative number to that power results in a positive number. This is because multiplying a negative number by itself an even number of times cancels out the negative signs. For instance, (-3)^{2} = 9 .
  • **Odd Exponents:** Contrarily, if the exponent is odd, like 1, 3, 5, etc., the result of the power remains negative. This is due to the single unpaired negative sign remaining after the multiplication. For example, (-3)^{3} = -27 .
Grasping these concepts is critical for solving problems involving powers, as it impacts the sign of the result.
Algebraic Expressions
An algebraic expression is a combination of numbers, variables, exponents, and operations such as addition, subtraction, multiplication, and division. These expressions can represent real-world situations and abstract mathematical concepts.
In algebra, understanding how to manage expressions with exponents, particularly with variables, is fundamental. Consider the expression (-3)^{n} :
  • The variable `n` , which acts as the exponent, determines the power to which the base, -3, is raised.
  • The power impacts both the size and the sign of the result. Thus, knowing whether `n` is odd or even helps in predicting if the outcome is positive or negative.
  • Simplifying these expressions often requires recognizing patterns in exponents and applying rules consistently.
Therefore, mastering algebraic expressions, especially those with exponents, is essential for solving equations and understanding more complex algebraic relations.

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Most popular questions from this chapter

Divide If the divisor contains 2 or more terms, use long division. $$ \frac{3-7 x^{2}}{x-3} $$

See the Concept Checks in this section. $$5 a \cdot 6 a$$

$$ \left(25 y^{11 b}+5 y^{6 b}-20 y^{3 b}+100 y^{b}\right) \div 5 y^{b} $$

Multiply. See Section 5.1 $$ 3 x(2 x) $$

Divide If the divisor contains 2 or more terms, use long division. $$ \frac{x^{3}+6 x^{2}+18 x+27}{x+3} $$
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