- Statement (a) \(\left| x^2 \right| = x^2\) is True because \(x^2 \ge 0\) for all real \(x\).
- Statement (b) \(\left| x^3 \right| = x^3\) is False; for \(x = -1\), \(\left| (-1)^3 \right| = 1\) while \((-1)^3 = -1\).
- Statement (c) \(\left| x^3 \right| = x^2 |x|\) is True as \(\left| x^3 \right| = \left| x^2 \cdot x \right| = \left| x^2 \right| \cdot |x| = x^2 |x|\).
- Final classifications: a: \(\boxed{\text{True}}\), b: \(\boxed{\text{False}}\), c: \(\boxed{\text{True}}\).
Explanation
- Analyzing Absolute Value Properties
To classify each statement as true or false for all real numbers \(x\), we need to analyze the properties of the absolute value function. The absolute value of a real number \(a\), denoted by \(|a|\), is defined as \(a\) if \(a \ge 0\) and \(-a\) if \(a < 0\). We also recall the property that \(|ab| = |a||b|\) and that \(|a|^2 = a^2\) for any real number \(a\).
- Evaluating Statement (a): \(\left| x^2 \right| = x^2\)
The statement is \(\left| x^2 \right| = x^2\). For any real number \(x\), the square \(x^2\) is always non-negative (\(x^2 \ge 0\)). By the definition of the absolute value, if a number is non-negative, its absolute value is equal to the number itself. Thus, \(\left| x^2 \right| = x^2\) for all real \(x\). This statement is true.
- Evaluating Statement (b): \(\left| x^3 \right| = x^3\)
The statement is \(\left| x^3 \right| = x^3\). For this to be true for all real \(x\), \(x^3\) must always be non-negative. However, if \(x\) is a negative number, say \(x = -1\), then \(x^3 = (-1)^3 = -1\). The absolute value is \(\left| x^3 \right| = \left| -1 \right| = 1\). Since \(1
eq -1\), the statement is false. A counterexample is \(x = -1\).
- Evaluating Statement (c): \(\left| x^3 \right| = x^2 |x|\)
The statement is \(\left| x^3 \right| = x^2 |x|\). Using the absolute value property \(\left| ab \right| = |a||b|\), we can rewrite the left-hand side as \(\left| x^3 \right| = \left| x^2 \cdot x \right| = \left| x^2 \right| \cdot |x|\). As established in step 2, \(x^2\) is always non-negative, so \(\left| x^2 \right| = x^2\). Substituting this back gives \(\left| x^2 \right| \cdot |x| = x^2 |x|\). Since the left side equals the right side for all real \(x\), this statement is true.
Examples
Understanding absolute value properties is essential in fields like physics and engineering, where magnitudes of vectors or physical quantities are often calculated. For instance, when determining the distance traveled by a particle moving along a line, you might need to compute \(\sqrt{x^2}\) or relate the magnitude of a velocity vector \(|v|\) to its components. Knowing that \(|x^2| = x^2\) allows for the simplification of complex expressions involving forces or fields, helping in accurately modeling physical systems.