To simplify the expression
[tex]\[
\left(\frac{1}{2}\right)^2 - 6\left(2 - \frac{2}{3}\right),
\][/tex]
we will break it down into step-by-step calculations:
1. Calculate [tex]\(\left(\frac{1}{2}\right)^2 \)[/tex]:
[tex]\[
\left(\frac{1}{2}\right)^2 = \frac{1}{4}
\][/tex]
Thus, [tex]\(\left(\frac{1}{2}\right)^2 = 0.25\)[/tex].
2. Simplify the expression inside the parentheses:
[tex]\[
2 - \frac{2}{3}
\][/tex]
To subtract these fractions, we need a common denominator. The common denominator of 2 and [tex]\(\frac{2}{3}\)[/tex] is 3. Convert 2 to [tex]\(\frac{6}{3}\)[/tex]:
[tex]\[
2 = \frac{6}{3}
\][/tex]
Now, we can subtract the fractions:
[tex]\[
\frac{6}{3} - \frac{2}{3} = \frac{6 - 2}{3} = \frac{4}{3}
\][/tex]
Therefore, [tex]\(2 - \frac{2}{3} = 1.3333333333333335\)[/tex].
3. Multiply the result by -6:
[tex]\[
-6 \times \left(\frac{4}{3}\right)
\][/tex]
Multiply the fractions directly:
[tex]\[
-6 \times \frac{4}{3} = -\frac{24}{3} = -8
\][/tex]
Thus, [tex]\(-6 \left(2 - \frac{2}{3}\right) = -8\)[/tex].
4. Combine the results:
[tex]\[
\left(\frac{1}{2}\right)^2 - 6\left(2 - \frac{2}{3}\right) = 0.25 - 8
\][/tex]
Simplify:
[tex]\[
0.25 - 8 = -7.75
\][/tex]
Therefore, the simplified form of the expression is:
[tex]\[
-7.75
\][/tex]
