Evaluate the expression sqrt(2(1)/(4) x 1((1)/(3))2) + [3]3(3)/(8).

The first term is $\sqrt{2\frac{1}{4} \times 1\left(\frac{1}{3}\right)^2}$.
First, convert the mixed number to an improper fraction:

$$2\frac{1}{4} = \frac{(2 \times 4) + 1}{4} = \frac{8 + 1}{4} = \frac{9}{4}$$

Next, calculate the square of the fraction:

$$\left(\frac{1}{3}\right)^2 = \frac{1^2}{3^2} = \frac{1}{9}$$

Now, substitute these values back into the first term:

$$\sqrt{\frac{9}{4} \times 1 \times \frac{1}{9}}$$

Multiply the fractions under the square root:

$$\sqrt{\frac{9}{4} \times \frac{1}{9}} = \sqrt{\frac{9 \times 1}{4 \times 9}} = \sqrt{\frac{9}{36}}$$

Simplify the fraction:

$$\sqrt{\frac{9}{36}} = \sqrt{\frac{1}{4}}$$

Take the square root:

$$\sqrt{\frac{1}{4}} = \frac{\sqrt{1}}{\sqrt{4}} = \frac{1}{2}$$

The second term is $\sqrt[3]{3\frac{3}{8}}$.
First, convert the mixed number to an improper fraction:

$$3\frac{3}{8} = \frac{(3 \times 8) + 3}{8} = \frac{24 + 3}{8} = \frac{27}{8}$$

Now, take the cube root:

$$\sqrt[3]{\frac{27}{8}} = \frac{\sqrt[3]{27}}{\sqrt[3]{8}}$$

Calculate the cube roots:

$$\sqrt[3]{27} = 3 \quad (\text{since } 3 \times 3 \times 3 = 27)$$ $$\sqrt[3]{8} = 2 \quad (\text{since } 2 \times 2 \times 2 = 8)$$

So, the second term is:

$$\frac{3}{2}$$

The total value of the expression is the sum of the two terms:

$$\frac{1}{2} + \frac{3}{2}$$

Add the fractions:

$$\frac{1+3}{2} = \frac{4}{2}$$

Simplify the result:

$$\frac{4}{2} = 2$$

The value of the expression is 2.