To simplify the expression \( \left(\frac{5}{3}\right)^{-3} \), follow these steps:

### Step 1: Apply the Negative Exponent Rule
The negative exponent rule states:
\[
a^{-n} = \frac{1}{a^n}.
\]
So:
\[
\left(\frac{5}{3}\right)^{-3} = \frac{1}{\left(\frac{5}{3}\right)^3}.
\]

### Step 2: Cube the Fraction
To cube the fraction \( \frac{5}{3} \), raise both the numerator and denominator to the power of 3:
\[
\left(\frac{5}{3}\right)^3 = \frac{5^3}{3^3} = \frac{125}{27}.
\]

### Step 3: Take the Reciprocal
The reciprocal of \( \frac{125}{27} \) is:
\[
\frac{1}{\frac{125}{27}} = \frac{27}{125}.
\]

### Final Answer:
\[
\left(\frac{5}{3}\right)^{-3} = \frac{27}{125}.
\]

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Would you like further clarification or details? Here are some related questions:

1. How do we simplify more complex fractions with exponents?
2. What happens when the base is negative in similar expressions?
3. Can you explain the general properties of exponents?
4. How does this process change if the base is a decimal instead of a fraction?
5. What is the connection between fractional exponents and roots?

**Tip:** When simplifying expressions with exponents, always check for rules like negative exponents or fractional bases to decide the correct approach.

Learn how to simplify the expression \( \left(\frac{5}{3}\right)^{-3} \) using the negative exponent rule and fraction operations. This step-by-step guide explains key concepts like exponentiation of fractions and the reciprocal property. Perfect for students in Grades 8-10.

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