Let's break down the expression step by step:

The expression is:

\[
\left(\frac{1}{2}\right)^3 + \left(\frac{1}{3}\right)^3 - \left(\frac{5}{6}\right)^3
\]

### Step 1: Calculate each cube.
- \(\left(\frac{1}{2}\right)^3 = \frac{1}{8}\)
- \(\left(\frac{1}{3}\right)^3 = \frac{1}{27}\)
- \(\left(\frac{5}{6}\right)^3 = \frac{125}{216}\)

### Step 2: Combine the fractions.
We need a common denominator for \(\frac{1}{8}\), \(\frac{1}{27}\), and \(\frac{125}{216}\).

The least common denominator (LCD) of 8, 27, and 216 is 216.

- \(\frac{1}{8} = \frac{27}{216}\)
- \(\frac{1}{27} = \frac{8}{216}\)
- \(\frac{125}{216}\) remains the same.

### Step 3: Substitute the fractions into the expression.

\[
\frac{27}{216} + \frac{8}{216} - \frac{125}{216}
\]

### Step 4: Combine the numerators.

\[
\frac{27 + 8 - 125}{216} = \frac{-90}{216}
\]

### Step 5: Simplify the fraction.

\[
\frac{-90}{216} = \frac{-15}{36} = \frac{-5}{12}
\]

Thus, the final answer is:

\[
\boxed{\frac{-5}{12}}
\]

Would you like more details or further explanations?

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Here are 5 related questions to expand your understanding:

1. What is the process to find the least common denominator (LCD) of three fractions?
2. How would the answer change if we had \((1/2)^4 + (1/3)^4 - (5/6)^4\)?
3. Can you calculate \(\left(\frac{2}{3}\right)^3 + \left(\frac{3}{4}\right)^3 - \left(\frac{1}{2}\right)^3\)?
4. How do you simplify negative fractions?
5. What happens if we add \(\frac{5}{12}\) instead of subtracting in the original expression?

**Tip:** When adding or subtracting fractions, always simplify the result by dividing both the numerator and denominator by their greatest common divisor (GCD).

Learn how to solve the expression (1/2)^3 + (1/3)^3 - (5/6)^3 by calculating the cubes of fractions, finding the least common denominator, and simplifying the result. This problem explores key mathematical concepts such as exponents, fractions, and simplification. Suitable for Grades 6-8 students.

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