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) $$ |x|=\sqrt{x^{2}}, x \in R $$

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**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).
Statements (a) and (b) are false.

Question 1 (1 point)
From the following statements, which one is false?
a)
b)
c)
d)

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