Question

How do you find the square root of $\dfrac{16}{121}$? \[\]

Answer 1

Hint: We recall the definition of square root and square root of a rational number. We recall that the square root of a rational number $\dfrac{a}{b}$ is a rational number $\dfrac{\sqrt{a}}{\sqrt{b}}$. We take $a=16,b=121$ and then find the required square root. \[\]

Complete step by step answer:
We know that the square root of the number $n$ is a number $a$ which multiplying with itself results in the product$n$. It means
\[a\times a=n\Rightarrow {{a}^{2}}=n\]
The square root is convent ally denoted as $\sqrt{n}=a$. The positive integers whose square root is a positive number are called perfect squares for example$1,4,9,16,...$. The square root of negative numbers does not exist. \[\]
We also know that the square root of a rational number $\dfrac{a}{b}$ whose denominator and numerator are both positive $a > 0 ,b >0 $ is given by the rational number whose numerator is square root of the numerator $a$ and denominator is square root of denominator $b$. It means
\[\sqrt{\dfrac{a}{b}}=\dfrac{\sqrt{a}}{\sqrt{b}}\]
We are asked in the question to find the square root of $\dfrac{16}{121}$. We see that the numerator is 16 and the denominator is 121. We see that both numerator and denominator are perfect squares since $16=4\times 4={{4}^{2}},121=11\times 11={{11}^{2}}$. So the square roots of numerator and denominator as follows
\[\sqrt{16}=4,\sqrt{121}=11\]

So the square root of given rational number is
\[\sqrt{\dfrac{16}{121}}=\dfrac{\sqrt{16}}{\sqrt{121}}=\dfrac{4}{11}\]

Note: We note the square root it means positive square root unless otherwise mentioned. The negative square root of $\dfrac{16}{121}$ is $-\dfrac{4}{11}$. We note that the square root of a rational number is a rational number when both numerator and denominator are perfect squares. We can find the square root of a rational number if one of the numerator or denominator is negative.