The problem provided in Italian translates to: "Simplify the following expressions by applying the properties of exponents."

The expression to simplify is: [ 1 + \left[ \left( \frac{3}{2} \right)^3 \right]^2 : \left{ \left[ \left( \frac{3}{2} \right)^2 \right]^1 : \left( -1 + \frac{1}{3} \right)^4 \right} ]

To solve this, we’ll go through each part, applying the properties of exponents and simplifying step-by-step.

Step-by-Step Solution

1. Simplify each exponent individually:

   • The term $\left( \frac{3}{2} \right)^3$ is raised to the power of 2. Using the property $(a^m)^n = a^{m \cdot n}$: $\left[ \left( \frac{3}{2} \right)^3 \right]^2 = \left( \frac{3}{2} \right)^{3 \cdot 2} = \left( \frac{3}{2} \right)^6$

   • Similarly, $\left( \frac{3}{2} \right)^2$ raised to the power of 1 remains the same: $\left[ \left( \frac{3}{2} \right)^2 \right]^1 = \left( \frac{3}{2} \right)^2$

2. Simplify the inner expression:

   • The inner term $\left( -1 + \frac{1}{3} \right)^4$ requires simplifying $-1 + \frac{1}{3}$: $-1 + \frac{1}{3} = -\frac{3}{3} + \frac{1}{3} = -\frac{2}{3}$ So, $\left( -1 + \frac{1}{3} \right)^4 = \left( -\frac{2}{3} \right)^4$
   • Raising $-\frac{2}{3}$ to the 4th power gives: $\left( -\frac{2}{3} \right)^4 = \frac{16}{81}$

3. Simplify the division in the brackets:

   • Now we have: $\frac{\left( \frac{3}{2} \right)^6}{\left( \frac{3}{2} \right)^2} : \frac{1}{\frac{16}{81}}$
   • Using the property $\frac{a^m}{a^n} = a^{m - n}$ for the fraction: $\frac{\left( \frac{3}{2} \right)^6}{\left( \frac{3}{2} \right)^2} = \left( \frac{3}{2} \right)^{6 - 2} = \left( \frac{3}{2} \right)^4$
   • Dividing by $\frac{16}{81}$ is equivalent to multiplying by $\frac{81}{16}$: $\left( \frac{3}{2} \right)^4 \cdot \frac{81}{16}$

4. Compute $\left( \frac{3}{2} \right)^4$: $\left( \frac{3}{2} \right)^4 = \frac{81}{16}$ So, $\frac{81}{16} \cdot \frac{81}{16} = \frac{6561}{256}$

5. Add 1: Finally, we have: $1 + \frac{6561}{256} = \frac{256 + 6561}{256} = \frac{6817}{256}$

Final Answer

$\frac{6817}{256}$

Would you like further details or have any questions on specific steps?

Related Questions

1. How do properties of exponents help in simplifying complex expressions?
2. What are the steps for converting complex fractions to simpler forms?
3. How do you handle negative bases raised to even powers?
4. What is the impact of raising a fraction to a high power on its overall value?
5. How can simplifying powers help solve equations more efficiently?

Tip

Always remember to simplify expressions inside parentheses or brackets first before applying exponent rules.