If x^(2), `4` and `9` are in continued proprotion, find `x`.

Answer

To solve the problem, we need to determine the value of \( x \) given that \( x^2 \), \( 4 \), and \( 9 \) are in continued proportion.

### Step-by-Step Solution:

1. **Understanding Continued Proportion**:
If three quantities \( a \), \( b \), and \( c \) are in continued proportion, then the relationship can be expressed as:
\[ \frac{a}{b} = \frac{b}{c} \]
This implies that the product of the extremes equals the product of the means:
\[ a \cdot c = b^2 \]

2. **Identifying the Terms**:
Here, we have:
- \( a = x^2 \)
- \( b = 4 \)
- \( c = 9 \)

3. **Setting Up the Equation**:
According to the property of continued proportion:
\[ x^2 \cdot 9 = 4^2 \]

4. **Calculating \( 4^2 \)**:
We know that:
\[ 4^2 = 16 \]
So, we can rewrite the equation as:
\[ x^2 \cdot 9 = 16 \]

5. **Solving for \( x^2 \)**:
To isolate \( x^2 \), divide both sides by \( 9 \):
\[ x^2 = \frac{16}{9} \]

6. **Finding \( x \)**:
To find \( x \), take the square root of both sides:
\[ x = \sqrt{\frac{16}{9}} = \frac{\sqrt{16}}{\sqrt{9}} = \frac{4}{3} \]

### Final Answer:
Thus, the value of \( x \) is:
\[ x = \frac{4}{3} \]