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Chapter 0: Problem 108

If \(x<0\) and \(m \in \mathbb{Z},\) can \(x^{m}\) be negative? If so, give an example.

Short Answer

Expert verified
Yes, \(x^{m}\) can be negative when \(m\) is an odd integer. For example, \( (-2)^{3} = -8\).

Step by step solution

01

Understand the given conditions

The exercise states that we need to find out if the expression \(x^{m}\) can be negative when \(x < 0\) and \(m\) is an integer.
02

Test with an odd integer exponent

Consider \(m\) to be an odd integer (e.g., \(1, 3, 5, ...\)). Let's choose \(m = 3\) and test the expression \(x^{m}\) with \(x = -2\). Calculate \( (-2)^{3} = -2 \times -2 \times -2 = -8 \). This result is negative.
03

Conclusion for odd exponents

From the previous step, it is clear that when \(m \) is an odd integer, \( x^{m} \) can indeed be negative if \( x \) is negative.
04

Example with different values

Try different values to further illustrate. For example, let \( x = -1\) and \( m = 5\). The calculation is \( (-1)^{5} = -1\), which is again negative.

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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 usually preceded by a minus sign (-). For example, -3 and -8 are negative numbers. They are important because they show values below zero, like temperatures or financial debts.

When you multiply or divide negative numbers, the rules are simple:
  • Negative 脳 Negative = Positive
  • Negative 脳 Positive = Negative
  • Positive 脳 Positive = Positive


    • Learning these rules can help make sense of operations involving negative values in algebraic expressions.
integer exponents
Integer exponents are exponents that are whole numbers, which can be positive, negative, or zero. If you have a number, say \(a\), raised to an integer exponent \(m\), it is written as \(a^{m}\). Here are some key points:
  • If \(m\) is positive, you multiply \(a\) by itself \(m\) times: for example, \(2^{3} = 2 \times 2 \times 2 = 8\).
  • If \(m\) is zero, any non-zero number raised to zero is \(1\): for example, \(2^{0} = 1\).
  • If \(m\) is negative, raise the number to the given positive exponent and then take the reciprocal: for example, \(2^{-3} = \frac{1}{2^{3}} = \frac{1}{8}\).

Understanding integer exponents is crucial, as they often appear in algebraic expressions and play a big role in solving equations.
odd exponents
Odd exponents are exponents that are odd integers, like 1, 3, 5, etc. When you raise a number to an odd exponent, certain rules apply:
  • If the base \(x\) is positive and the exponent \(m\) is odd, then the result \(x^{m}\) will be positive.
  • If the base \(x\) is negative and the exponent \(m\) is odd, then the result \(x^{m}\) will be negative.
For example, \((-2)^{3} = -2 \times -2 \times -2 = -8\). You can see that raising a negative number to an odd exponent keeps the result negative.

This highlights an important pattern in algebra and helps predict the signs of results in calculations.
algebraic expressions
Algebraic expressions are combinations of variables, numbers, and operations like addition, multiplication, and exponents. For instance, \(2x + 3\) and \(x^{2} - 4x + 5\) are algebraic expressions.

When dealing with algebraic expressions involving exponents:
  • Keep in mind the rules for negative numbers and exponents.
  • Pay attention to the signs and operations to simplify or solve the expression.
  • Use the power of algebraic identities to factorize or expand expressions when needed.
Understanding algebraic expressions is foundational for solving more complex math problems and working through equations effectively.

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Most popular questions from this chapter

Multiply and simplify. Assume that all variable expressions represent positive real numbers. $$ 4 \sqrt{10}(7 \sqrt{2}-3 \sqrt{5}+\sqrt{10}) $$

Perform the indicated operations and simplify. $$ (7 v-2)^{2} $$

Multiply and simplify. Assume that all variable expressions represent positive real numbers. $$ (\sqrt{7+3 \sqrt{z}})(\sqrt{7-3 \sqrt{z}}) $$

Add or subtract as indicated and simplify. $$ \left(-8 p^{7}-4 p^{4}+2 p-5\right)+\left(2 p^{7}+6 p^{4}+p^{2}\right) $$

If \(n<0,\) then \(n-|n|=\)_____.
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