Let's break down and simplify the given expression step by step.
The given expression is:
[tex]\[
\left(\frac{1}{2}\right)^2 - 6 \left(2 - \frac{2}{3}\right)
\][/tex]
### Step 1: Simplify [tex]\(\left(\frac{1}{2}\right)^2\)[/tex]
First, calculate [tex]\(\left(\frac{1}{2}\right)^2\)[/tex]:
[tex]\[
\left(\frac{1}{2}\right)^2 = \frac{1}{4}
\][/tex]
### Step 2: Simplify [tex]\(2 - \frac{2}{3}\)[/tex]
Next, simplify the expression inside the parentheses:
[tex]\[
2 - \frac{2}{3}
\][/tex]
To subtract these fractions, let's convert 2 to a fraction with a common denominator of 3:
[tex]\[
2 = \frac{6}{3}
\][/tex]
Now subtract:
[tex]\[
\frac{6}{3} - \frac{2}{3} = \frac{4}{3}
\][/tex]
### Step 3: Multiply by 6
Next, multiply the result by 6:
[tex]\[
6 \left(\frac{4}{3}\right) = \frac{24}{3} = 8
\][/tex]
### Step 4: Combine the results
Now combine all the simplified parts:
[tex]\[
\left(\frac{1}{4}\right) - 8
\][/tex]
To combine these terms, express [tex]\(8\)[/tex] as a fraction with a common denominator of 4:
[tex]\[
8 = \frac{32}{4}
\][/tex]
Subtract the fractions:
[tex]\[
\frac{1}{4} - \frac{32}{4} = \frac{1 - 32}{4} = \frac{-31}{4}
\][/tex]
Thus, the simplified expression is:
[tex]\[
\boxed{\frac{-31}{4}}
\][/tex]
