To simplify the expression [tex]\(\frac{1}{4}\left(1-\frac{2}{3}\right)^2+\frac{1}{3}\)[/tex], let's go through it step by step.
1. Calculate the value inside the parentheses:
[tex]\[
1 - \frac{2}{3}
\][/tex]
To subtract these fractions, convert the whole number 1 into a fraction with the same denominator as [tex]\(\frac{2}{3}\)[/tex].
[tex]\[
1 = \frac{3}{3}
\][/tex]
Now, subtract:
[tex]\[
\frac{3}{3} - \frac{2}{3} = \frac{1}{3}
\][/tex]
So, the value inside the parentheses is [tex]\(\frac{1}{3}\)[/tex].
2. Square this result:
[tex]\[
\left(\frac{1}{3}\right)^2 = \frac{1}{9}
\][/tex]
3. Multiply this squared result by the fraction [tex]\(\frac{1}{4}\)[/tex]:
[tex]\[
\frac{1}{4} \cdot \frac{1}{9}
\][/tex]
To multiply two fractions, multiply the numerators together and the denominators together:
[tex]\[
\frac{1 \cdot 1}{4 \cdot 9} = \frac{1}{36}
\][/tex]
4. Add the second fraction [tex]\(\frac{1}{3}\)[/tex] to this result:
To add [tex]\(\frac{1}{36}\)[/tex] and [tex]\(\frac{1}{3}\)[/tex], we need a common denominator. The least common multiple of 36 and 3 is 36. Convert [tex]\(\frac{1}{3}\)[/tex] to a fraction with a denominator of 36:
[tex]\[
\frac{1}{3} = \frac{12}{36}
\][/tex]
So now, add [tex]\(\frac{1}{36}\)[/tex] and [tex]\(\frac{12}{36}\)[/tex]:
[tex]\[
\frac{1}{36} + \frac{12}{36} = \frac{1 + 12}{36} = \frac{13}{36}
\][/tex]
Therefore, the simplified result of the expression [tex]\(\frac{1}{4}\left(1-\frac{2}{3}\right)^2+\frac{1}{3}\)[/tex] is:
[tex]\[
\boxed{\frac{13}{36}}
\][/tex]
