Question

Find the square root of \[\dfrac{289}{144}\] by prime factorisation method.

Answer 1

Hint: Do the prime factorisation of 289 and 144 separately , Substitute those back in the question given , if any terms gets cancelled then fine , otherwise consider the pair which is repeated as one number in both numerator and denominator for square root.

Complete step-by-step answer:
From the question, it is clear that we should find the square root of \[\dfrac{289}{144}\].
So, first we should find the square root of 289.
Now we should write 289 in a perfect square.
Now we have to find the factors of 289 by prime factorisation.
\[\begin{align}
  & 289=1\times 289 \\
 & 289=17\times 17 \\
 & 289=289\times 1 \\
\end{align}\]
So, we can say that the 17 is the prime factor of 289.
Hence, we can write 289 as the perfect square of 17 as shown below.
\[\begin{align}
  & \Rightarrow 289=17\times 17 \\
 & \Rightarrow 289={{\left( 17 \right)}^{2}}....(1) \\
\end{align}\]
Now we have to find the factors of 144 by prime factorisation.
\[\begin{align}
  & 144=1\times 144 \\
 & 144=2\times 72 \\
 & 144=3\times 48 \\
 & 144=4\times 36 \\
 & 144=6\times 24 \\
 & 144=8\times 18 \\
 & 144=12\times 12 \\
\end{align}\]
So, we can say that 144 can be written as a perfect square of 12 as shown below.
\[\begin{align}
  & \Rightarrow 144=12\times 12 \\
 & \Rightarrow 144={{\left( 12 \right)}^{2}}....(2) \\
\end{align}\]
Now we have to find the square root of \[\dfrac{289}{144}\].
Let us assume the square root of \[\dfrac{289}{144}\] is equal to A.
\[\Rightarrow A=\sqrt{\dfrac{289}{144}}.....(3)\]
We know that \[\sqrt{\dfrac{a}{b}}\] can be written as \[\dfrac{\sqrt{a}}{\sqrt{b}}\].
So, from equation (3) we get
\[\Rightarrow \sqrt{\dfrac{289}{144}}=\dfrac{\sqrt{289}}{\sqrt{144}}\]
Now from equation (1) and equation (2), we get
\[\Rightarrow \dfrac{\sqrt{289}}{\sqrt{144}}=\dfrac{\sqrt{{{\left( 17 \right)}^{2}}}}{\sqrt{{{\left( 12 \right)}^{2}}}}\]
We know that the value of \[\sqrt{{{a}^{2}}}\] is equal to a.
\[\Rightarrow \dfrac{\sqrt{{{\left( 17 \right)}^{2}}}}{\sqrt{{{\left( 12 \right)}^{2}}}}=\dfrac{17}{12}\]
So, it is clear that the square root of \[\dfrac{289}{144}\] is equal to \[\dfrac{17}{12}\].

Note: Students should write the value of 289 and 144 in the form of perfect square. Otherwise, it is impossible to find the value square root of 289 and 144. Then we cannot find the square root of \[\dfrac{289}{144}\]. So, we should find the perfect square. Then we should find the square root of 289 and 144. Students should also take care of the calculation part. If a small mistake is made, we cannot get the accurate answer. So, calculation should be done in a perfect manner.