Simplify:
`1/4(1−2/3)2+1/3`

Answer

To simplify the expression \( \frac{1}{4}(1 - \frac{2}{3})^2 + \frac{1}{3} \), we will follow these steps:

### Step 1: Simplify the expression inside the parentheses
We start with the expression inside the parentheses:
\[ 1 - \frac{2}{3} \]
To subtract these, we need a common denominator. The common denominator for 1 (which can be written as \( \frac{3}{3} \)) and \( \frac{2}{3} \) is 3.

\[ 1 = \frac{3}{3} \]
So,
\[ 1 - \frac{2}{3} = \frac{3}{3} - \frac{2}{3} = \frac{1}{3} \]

### Step 2: Substitute back into the expression
Now we substitute \( \frac{1}{3} \) back into the expression:
\[ \frac{1}{4} \left( \frac{1}{3} \right)^2 + \frac{1}{3} \]

### Step 3: Calculate \( \left( \frac{1}{3} \right)^2 \)
Now we calculate \( \left( \frac{1}{3} \right)^2 \):
\[ \left( \frac{1}{3} \right)^2 = \frac{1^2}{3^2} = \frac{1}{9} \]

### Step 4: Substitute and multiply by \( \frac{1}{4} \)
Now we substitute \( \frac{1}{9} \) back into the expression:
\[ \frac{1}{4} \cdot \frac{1}{9} + \frac{1}{3} \]
Now, multiply \( \frac{1}{4} \) and \( \frac{1}{9} \):
\[ \frac{1}{4} \cdot \frac{1}{9} = \frac{1 \cdot 1}{4 \cdot 9} = \frac{1}{36} \]

### Step 5: Add \( \frac{1}{36} \) and \( \frac{1}{3} \)
Next, we need to add \( \frac{1}{36} \) and \( \frac{1}{3} \). To do this, we need a common denominator. The least common multiple of 36 and 3 is 36.

Convert \( \frac{1}{3} \) to have a denominator of 36:
\[ \frac{1}{3} = \frac{12}{36} \]

Now we can add:
\[ \frac{1}{36} + \frac{12}{36} = \frac{1 + 12}{36} = \frac{13}{36} \]

### Final Answer
Thus, the simplified expression is:
\[ \frac{13}{36} \]

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