Which of the following conditions necessarily imply that the real number x is rational, I `x^2` is rational II `x^3`and `x^5`are rational III `x^2`and `x^3` are rational

Options

A. I and II only
B. I and III only
C. II and III only
D. III only

Answer

To determine which of the given conditions necessarily imply that the real number \( x \) is rational, we will analyze each condition step by step.

### Step 1: Analyze Condition I
**Condition I:** \( x^2 \) is rational.

- Suppose \( x^2 \) is rational. This means that there exists some rational number \( r \) such that \( x^2 = r \).
- However, \( x \) could be either \( \sqrt{r} \) or \( -\sqrt{r} \). If \( r \) is a rational number, then both \( \sqrt{r} \) and \( -\sqrt{r} \) could be either rational or irrational depending on \( r \).
- For example, if \( r = 2 \), then \( x = \sqrt{2} \) (which is irrational).

**Conclusion for Condition I:** This condition does not necessarily imply that \( x \) is rational.

### Step 2: Analyze Condition II
**Condition II:** \( x^3 \) and \( x^5 \) are rational.

- Let \( x^3 = a \) and \( x^5 = b \), where both \( a \) and \( b \) are rational.
- We can express \( x \) in terms of \( x^3 \) and \( x^5 \):
\[ x^5 = x^3 \cdot x^2 \implies x^2 = \frac{x^5}{x^3} = \frac{b}{a} \]
- Since both \( a \) and \( b \) are rational, \( \frac{b}{a} \) is also rational, which implies that \( x^2 \) is rational.
- If \( x^2 \) is rational, then \( x \) must be rational (as established in Condition I).

**Conclusion for Condition II:** This condition does imply that \( x \) is rational.

### Step 3: Analyze Condition III
**Condition III:** \( x^2 \) and \( x^3 \) are rational.

- Let \( x^2 = m \) and \( x^3 = n \), where both \( m \) and \( n \) are rational.
- From \( x^3 = x^2 \cdot x \), we have:
\[ x = \frac{x^3}{x^2} = \frac{n}{m} \]
- Since both \( n \) and \( m \) are rational, \( \frac{n}{m} \) is also rational, which implies that \( x \) is rational.

**Conclusion for Condition III:** This condition also implies that \( x \) is rational.

### Final Conclusion
The conditions that necessarily imply that the real number \( x \) is rational are:
- Condition II: \( x^3 \) and \( x^5 \) are rational.
- Condition III: \( x^2 \) and \( x^3 \) are rational.

### Summary of Results
- Condition I does not imply \( x \) is rational.
- Condition II implies \( x \) is rational.
- Condition III implies \( x \) is rational.