Answered

Classify each of the following as true or false for all values of $x$ and $y$. If false, show a counterexample that makes it false.


A$\left| x ^2\right|= x ^2$
B$\left|x^3\right|=x^3$
C$\left|x^3\right|=x^2|x|$


ASelect the correct choice below.
AThe statement $\left|x^2\right|=x^2$ is false. A counterexample is $x=0$.
BThe statement $\left|x^2\right|=x^2$ is true.
CThe statement $\left|x^2\right|=x^2$ is false. A counterexample is $x=-1$.
DThe statement $\left|x^2\right|=x^2$ is false. A counterexample is $x=1$.


BSelect the correct choice below.
AThe statement $\left|x^3\right|=x^3$ is false. A counterexample is $x=1$.
BThe statement $\left|x^3\right|=x^3$ is false. A counterexample is $x=0$.
CThe statement $\left|x^3\right|=x^3$ is false. A counterexample is $x=-1$.

Verified Answer :

  • Statement (a) \(\left| x^2 \right| = x^2\) is true because \(x^2\) is always non-negative.
  • Statement (b) \(\left| x^3 \right| = x^3\) is false; a counterexample is \(x = -1\), where \(|(-1)^3| = 1 eq -1\).
  • Statement (c) \(\left| x^3 \right| = x^2 |x|\) is true by property \(\left| x^3 \right| = \left| x^2 \right| \cdot \left| x \right| = x^2 |x|\).
  • Answers: a) B, b) C, c) True.

Explanation

  1. Problem Analysis and Definitions

The goal is to analyze the truth value of three specific mathematical statements involving the absolute value function and powers of a real number \(x\). The absolute value of a real number \(a\), denoted by \(\left| a \right|\), is defined as \(a\) if \(a \geq 0\) and \(-a\) if \(a < 0\). A fundamental property of absolute values is that for any real numbers \(a\) and \(b\), \(\left| a \cdot b \right| = \left| a \right| \cdot \left| b \right|\). Additionally, \(\left| a^n \right| = \left| a \right|^n\) for any integer \(n\). We will examine each statement sequentially.

  1. Analyzing Part (a)

For statement (a), we examine \(\left| x^2 \right| = x^2\). For any real number \(x\), the square \(x^2\) is always greater than or equal to zero (\(x^2 \geq 0\)). By the definition of the absolute value, if \(a \geq 0\), then \(\left| a \right| = a\). Letting \(a = x^2\), we have \(\left| x^2 \right| = x^2\). Therefore, the statement is true for all real \(x\). This corresponds to option B.

  1. Analyzing Part (b)

For statement (b), we examine \(\left| x^3 \right| = x^3\). If \(x\) is a positive number, the equality holds. However, if \(x\) is negative, \(x^3\) is also negative. For example, let \(x = -1\). Then \(\left| x^3 \right| = \left| (-1)^3 \right| = \left| -1 \right| = 1\), whereas \(x^3 = (-1)^3 = -1\). Since \(1 eq -1\), the statement \(\left| x^3 \right| = x^3\) is false. The counterexample \(x = -1\) proves this, matching option C.

  1. Analyzing Part (c)

For statement (c), we examine \(\left| x^3 \right| = x^2 |x|\). Using the properties of absolute values, we can write \(\left| x^3 \right| = \left| x^2 \cdot x \right| = \left| x^2 \right| \cdot \left| x \right|\). As established in the analysis for part (a), \(\left| x^2 \right| = x^2\) because \(x^2\) is always non-negative. Therefore, \(\left| x^3 \right| = x^2 \cdot \left| x \right|\). This demonstrates that the statement is true for all real \(x\).

Examples

Absolute value properties are essential in fields like physics and engineering when measuring magnitudes of quantities that can be positive or negative, such as displacement, velocity, or electric charge. For instance, if you are calculating the total distance traveled by an object moving back and forth along a line, you use absolute values to ensure each segment of the movement contributes positively to the total distance. Understanding these algebraic identities allows for simpler expressions when modeling complex physical systems and ensures calculations remain consistent regardless of the sign of the input variables.

Related Questions