To do \(\left(\frac{1}{2}\frac{2}{5}\right)^2\), we will first have to convert both fractions into fractions that have common denominators. To do this, we must figure out the LCM (lowest common multiple) of both denominators in the expression. In this example, 2 and 5 are coprime, so simply multiply them together to figure out the LCM. The LCM is 2*%, or 10. Let's convert both fractions such that both have a denominator of 10. I will manipulate the fractions chronologically in which they appear in the expression:
\(\frac{1}{2}*\frac{5}{5}\)  Note that we are really multiplying the fraction by 1, so the actual value of the fraction will be unchanged. 
\(\frac{5}{10}\)  
And of course, the next fraction:
\(\frac{2}{5}*\frac{2}{2}\)  This accomplishes the exact same thing as abovegetting a common denominator. 
\(\frac{4}{10}\)  
Let's subtract the fractions, now:
\(\left(\frac{5}{10}\frac{4}{10}\right)^2\)  Subtracting the fraction involves subtracting the numerator and keeping the denominator unchanged. 
\(\left(\frac{1}{10}\right)^2\)  To square a fraction, use the following rule of \(\left(\frac{a}{b}\right)^n=\frac{a^n}{b^n}\). You are "distributing" the fraction to both the numerator and denominator. 
\(\frac{1^2}{10^2}\)  Simplify both the numerator and denominator. 
\(\frac{1}{100}\) 

Therefore, \(\left(\frac{1}{2}\frac{2}{5}\right)^2=\frac{1}{100}\)