To solve $\left(\frac{3}{2}\right)^{-3}$, follow these steps:

Step 1: Apply the Negative Exponent Rule

The negative exponent rule states that $a^{-n} = \frac{1}{a^n}$. So, we apply this to the expression:

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

Step 2: Calculate the Cube of $\frac{3}{2}$

Now, we need to raise $\frac{3}{2}$ to the power of 3:

$\left(\frac{3}{2}\right)^3 = \frac{3^3}{2^3} = \frac{27}{8}$

Step 3: Substitute Back

Now, substitute $\frac{27}{8}$ into the expression:

$\left(\frac{3}{2}\right)^{-3} = \frac{1}{\frac{27}{8}} = \frac{8}{27}$

Final Answer:

$\left(\frac{3}{2}\right)^{-3} = \frac{8}{27}$

Would you like further clarification or details on any step?

Related Questions:

1. What happens when we apply positive exponents to fractions like $\left(\frac{3}{2}\right)^3$?
2. How can the laws of exponents be used to simplify expressions with negative exponents?
3. How do you simplify expressions like $\left(\frac{5}{3}\right)^{-2}$?
4. How would you handle fractional exponents, for example $\left(\frac{3}{2}\right)^{1/2}$?
5. Can negative exponents be simplified differently when applied to whole numbers?

Tip: Remember, negative exponents always mean "take the reciprocal" of the base before applying the exponent.