Irrational number: A number is said to be irrational if it can't be written in the form of p/q, q≠0, the decimal expansion is non terminating non repeating.
Also a fraction is rational if it can be written in the form of p/q,q≠0, also it's decimal terminates if it's denominator consists of [tex]2^{n}, 5^{m} or 2^{n}\times 5^{m}[/tex].
1. [tex]\frac{3}{16}[/tex]= [tex]\frac{3}{2^{4}}[/tex] =Rational number
2. [tex]\sqrt{\frac{4}{16}}= \sqrt{\frac{1}{4}}= \frac{1}{2}[/tex] = Rational number
3 . [tex]\sqrt{\frac{9}{16}}= \sqrt{\frac{3^{2}} {4^{2}} }=\frac{3}{4}= \frac{3}{2^{2}}[/tex] = Rational number
4. [tex]\sqrt{\frac{3}{4}}\times \sqrt{\frac{9}{4}}= \frac {\sqrt{27} }{4}=\frac{3 \sqrt{3} }{4}[/tex] = As √3 is irrational and 3/4 is rational so product of rational and irrational is irrational.
So, option (4) which is [tex]\sqrt{\frac{3}{4}}\times \sqrt{\frac{9}{4}}[/tex] is irrational.
