Question 1 (1 point) From the following statements, which one is false? a) \( |x|=\sqrt[3]{x}, x \in R \) b) \( x=\sqrt[3]{x}, x \geq 0 \) c) \( x=\sqrt{x^{2}}, x \geq 0 \) d) \( |x|=\sqrt{x^{2}}, x \in R \)

**Step 1. Verify Statement (a):** The statement is \[ |x|=\sqrt[3]{x} \quad \text{for } x\in \mathbb{R}. \] Recall that the cube root function, \(\sqrt[3]{x}\), is defined for all real numbers and is an odd function. This means \[ \sqrt[3]{x} < 0 \quad \text{if } x < 0, \] whereas the absolute value \(|x|\) is always nonnegative. For a counter‐example, take \(x=-1\): - Left side: \(|-1| = 1\). - Right side: \(\sqrt[3]{-1}=-1\). Since \(1\neq -1\), statement (a) is false. --- **Step 2. Verify Statement (b):** The statement is \[ x=\sqrt[3]{x} \quad \text{for } x\ge0. \] For \(x\ge0\) the cube root is defined and is nonnegative, but we must check if the equality holds for every such \(x\). Test with \(x=8\): - Left side: \(8\). - Right side: \(\sqrt[3]{8}=2\). Since \(8\neq2\), statement (b) does not hold for all \(x\ge0\). Hence, statement (b) is also false. --- **Step 3. Verify Statement (c):** The statement is \[ x=\sqrt{x^2} \quad \text{for } x\ge0. \] For any real \(x\), we have the identity \[ \sqrt{x^2}=|x|. \] If \(x\ge0\), then \(|x|=x\). Thus, the equality holds for every \(x\ge0\), and statement (c) is true. --- **Step 4. Verify Statement (d):** The statement is \[ |x|=\sqrt{x^2} \quad \text{for } x\in\mathbb{R}. \] As noted, the identity \(\sqrt{x^2}=|x|\) holds true for all real numbers. Therefore, statement (d) is true. --- **Conclusion:** The false statements are (a) and (b).