Certainly! Let's solve the given expression step-by-step:
[tex]\[
\left(\frac{1}{2}\right)^2 - 6\left(2 - \frac{2}{3}\right)
\][/tex]
Step 1: Calculate [tex]\(\left(\frac{1}{2}\right)^2\)[/tex]:
[tex]\[
\left(\frac{1}{2}\right)^2 = \frac{1}{4} = 0.25
\][/tex]
Step 2: Simplify the expression inside the parentheses:
[tex]\[
2 - \frac{2}{3}
\][/tex]
To subtract, convert 2 into a fraction with a common denominator:
[tex]\[
2 = \frac{6}{3}
\][/tex]
So, the expression becomes:
[tex]\[
\frac{6}{3} - \frac{2}{3} = \frac{4}{3}
\][/tex]
Hence,
[tex]\[
2 - \frac{2}{3} = \frac{4}{3} \approx 1.3333333333333335
\][/tex]
Step 3: Multiply the simplified expression by -6:
[tex]\[
-6 \times \frac{4}{3} = -6 \times \left(\frac{4}{3}\right) = -\frac{24}{3} = -8
\][/tex]
Step 4: Add the results from steps 1 and 3:
[tex]\[
\left(\frac{1}{2}\right)^2 + \left( - 6 \left(2 - \frac{2}{3}\right) \right) = 0.25 + (-8)
\][/tex]
Thus,
[tex]\[
0.25 - 8 = -7.75
\][/tex]
Now, convert [tex]\(-7.75\)[/tex] into a fraction in simplest form. We know:
[tex]\[
-7.75 = -\frac{31}{4}
\][/tex]
Hence, the value of the expression [tex]\(\left(\frac{1}{2}\right)^2 - 6\left(2 - \frac{2}{3}\right)\)[/tex] as a fraction in simplest form is:
[tex]\[
\boxed{-\frac{31}{4}}
\][/tex]
