Question

If the Square root of $23\frac{394}{729}=\frac{a}{27}$, then find the value of $a$?

Answer 1

$23\frac{394}{729}$

$\sqrt{23\frac{394}{729}}$ $=\sqrt{\frac{17161}{729}}$ $=\frac{\sqrt{17161}}{\sqrt{729}}$

Now by using prime factorization method we are going to find square root of $17161$ and $729$ separately.

$17161=131\times 131$

$\Rightarrow$ $17161={131}^2$

$\Rightarrow$ $\sqrt{17161}=\sqrt{{131}^2}$

$\Rightarrow$ $\sqrt{17161}=131$ ----- ( 1 )

Now,

$729=3\times 243$

$\Rightarrow$ $729=3\times 3\times 81$

$\Rightarrow$ $729=3\times 3\times 3\times 27$

$\Rightarrow$ $729=3\times 3\times 3\times 3\times 9$

$\Rightarrow$ $729=3\times 3\times 3\times 3\times 3\times 3$

$\Rightarrow$ $729=3^6$

$\Rightarrow$ $\sqrt{729}=\sqrt{3^6}$

$\Rightarrow$ $\sqrt{729}=3^3$

$\Rightarrow$ $\sqrt{729}=27$ ---- ( 2 )

From ( 1 ) and ( 2 )

$\frac{\sqrt{17161}}{\sqrt{729}}=\frac{131}{27}$

$\therefore$ Square root of $23\frac{394}{729}$ is $\frac{131}{27}$