Solve the following expression: $$(2\frac{1}{2})^3 - 0.34 = $$ Solve the following expression and represent it as a fraction: $$\frac{11}{32} : 2\frac{5}{12} + 0.4 =$$

Let's solve the first expression step by step:

First, convert the mixed number to an improper fraction:

$$2\frac{1}{2} = \frac{2 \cdot 2 + 1}{2} = \frac{5}{2}$$

Now, raise the fraction to the power of 3:

$$(\frac{5}{2})^3 = \frac{5^3}{2^3} = \frac{125}{8}$$

Convert the decimal 0.34 to a fraction:

$$0.34 = \frac{34}{100} = \frac{17}{50}$$

Now, subtract the second fraction from the first:

$$\frac{125}{8} - \frac{17}{50}$$

Find a common denominator. The least common multiple of 8 and 50 is 200.

$$\frac{125}{8} = \frac{125 \cdot 25}{8 \cdot 25} = \frac{3125}{200}$$ $$\frac{17}{50} = \frac{17 \cdot 4}{50 \cdot 4} = \frac{68}{200}$$

Now, subtract:

$$\frac{3125}{200} - \frac{68}{200} = \frac{3125 - 68}{200} = \frac{3057}{200}$$

So, the first expression equals $$\frac{3057}{200}$$.

Now, let's solve the second expression step by step:

Convert the mixed number to an improper fraction:

$$2\frac{5}{12} = \frac{2 \cdot 12 + 5}{12} = \frac{29}{12}$$

Now we have:

$$\frac{11}{32} : \frac{29}{12} + 0.4$$

To divide by a fraction, we multiply by its reciprocal:

$$\frac{11}{32} \cdot \frac{12}{29} + 0.4$$

Multiply the fractions:

$$\frac{11 \cdot 12}{32 \cdot 29} = \frac{132}{928} = \frac{33}{232}$$

Convert the decimal 0.4 to a fraction:

$$0.4 = \frac{4}{10} = \frac{2}{5}$$

Now, add the fractions:

$$\frac{33}{232} + \frac{2}{5}$$

Find a common denominator. The least common multiple of 232 and 5 is 1160.

$$\frac{33}{232} = \frac{33 \cdot 5}{232 \cdot 5} = \frac{165}{1160}$$ $$\frac{2}{5} = \frac{2 \cdot 232}{5 \cdot 232} = \frac{464}{1160}$$

Now, add:

$$\frac{165}{1160} + \frac{464}{1160} = \frac{165 + 464}{1160} = \frac{629}{1160}$$

So, the second expression equals $$\frac{629}{1160}$$.

Answer:

First expression: $$\frac{3057}{200}$$

Second expression: $$\frac{629}{1160}$$