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Let's break down the given expression and solve it step by step. The expression is:

\[
\frac{1}{2} \left( \frac{5}{6} \right)^2 - \frac{1}{8}
\]

### Step 1: Square \(\frac{5}{6}\)
Squaring \(\frac{5}{6}\) gives:
\[
\left( \frac{5}{6} \right)^2 = \frac{25}{36}
\]

### Step 2: Multiply \(\frac{1}{2}\) by \(\frac{25}{36}\)
Now, multiply the two fractions:
\[
\frac{1}{2} \times \frac{25}{36} = \frac{25}{72}
\]

### Step 3: Subtract \(\frac{1}{8}\) from \(\frac{25}{72}\)
To subtract these fractions, we need a common denominator. The least common denominator (LCD) of 72 and 8 is 72. Convert \(\frac{1}{8}\) to have a denominator of 72:
\[
\frac{1}{8} = \frac{9}{72}
\]

Now subtract:
\[
\frac{25}{72} - \frac{9}{72} = \frac{16}{72}
\]

### Step 4: Simplify the result
\(\frac{16}{72}\) can be simplified by dividing both the numerator and denominator by their greatest common divisor, which is 8:
\[
\frac{16}{72} = \frac{2}{9}
\]

### Final Answer:
The simplified form of the expression is:
\[
\boxed{\frac{2}{9}}
\]

Would you like more details on any of the steps, or do you have any other questions?

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Here are five related questions:
1. How do you find the least common denominator for two fractions?
2. What is the process for multiplying two fractions?
3. How do you simplify a fraction to its lowest terms?
4. How do you solve expressions involving both exponents and fractions?
5. How do you subtract fractions with different denominators?

**Tip:** Always simplify fractions to their lowest terms by dividing both the numerator and denominator by their greatest common divisor (GCD).

Evaluate the expression 1/2 * (5/6)^2 - 1/8 and write the answer in simplest form.

Learn how to solve the expression 1/2 * (5/6)^2 - 1/8 with a detailed step-by-step breakdown. This problem covers key concepts like fraction multiplication, exponentiation, and subtraction of fractions with different denominators. Suitable for students in Grades 6-8.

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