Question

The square root of \[\dfrac{36}{5}\] correct to two decimal places is ____________.
A. 2.68
B. 2.69
C. 2.67
D. 2.66

Answer 1

Hint: First we have to calculate the prime factors of 36 by factorization method. Then we have to calculate the value of \[\sqrt{5}\] by a long division method, as we know 5 is not a perfect square. After getting the value of \[\sqrt{5}\] , we simply have to divide it with the square root of 36. After evaluating this way we can find the answer.

Complete step-by-step answer:
We know that a square root of a number is a value that, when multiplied by itself, gives the number. For example \[4\times 4=16\]. So a square root of 16 is 4.
Now we can write the prime factors under square root, as we have to evaluate the square root of that number.
But here we can only get prime factors from numerator which is
\[\begin{align}
  & 2\left| \!{\underline {\,
  36 \,}} \right. \\
 & 2\left| \!{\underline {\,
  18 \,}} \right. \\
 & 3\left| \!{\underline {\,
  6 \,}} \right. \\
 & 3\left| \!{\underline {\,
  3 \,}} \right. \\
 & \text{ }1 \\
\end{align}\]
So that,
 \[\begin{align}
  & \sqrt{36} \\
 & \Rightarrow \sqrt{2\times 2\times 3\times 3} \\
 & \Rightarrow \sqrt{{{2}^{2}}\times {{3}^{2}}} \\
 & \Rightarrow \sqrt{{{6}^{2}}} \\
 & \Rightarrow 6 \\
\end{align}\]
We don’t get the square root of 5 that easily. Value of \[\sqrt{5}\] could be evaluated, by a long division method, as we know 5 is not a perfect square. The steps for finding the value of\[\sqrt{5}\] are,
We have to write the number 5 as 5.00000000. that is,
\[5=5.00\text{ 00 00}\]
Then we have to take the number whose square is below 5. So, 4 is the perfect square of number 2 and it is below 5 and writes the number 2 in both divisor and quotient place, such that when 2 multiplied by 2 gives 4.
\[\begin{align}
  & \text{ }2.23 \\
 & \text{ }2\left| \!{\overline {\,
 5.00\text{ }00\text{ }00 \,}} \right. \\
 & \text{ }4 \\
\end{align}\]
Next we have to carry down two zeros and write down after 1 and take the decimal point after 1 in the quotient. Now add 2 in the divisor to make it 4. Take a number next to 4, such that when we multiply the combination with the same number, we get a number which is less than or equal to 100.

\[\begin{align}
  & \text{ }2.23 \\
 & \text{ }2\left| \!{\overline {\,
 5.00\text{ }00\text{ }00 \,}} \right. \\
 & \text{ }4 \\
 & \text{ }42\left| \!{\overline {\,
 100 \,}} \right. \\
 & \text{ -84} \\
 & 443\left| \!{\overline {\,
 1600 \,}} \right. \\
 & \text{ }-1329 \\
 & \text{ }\left| \!{\overline {\,
 271 \,}} \right. \\
\end{align}\]
Finally, we get the quotient value as 2.23.
Now we can evaluate \[\sqrt{\dfrac{36}{5}}\] as \[\sqrt{\dfrac{{{6}^{2}}}{{{\left( 2.23 \right)}^{2}}}}\].
The value of square root of \[\dfrac{36}{5}\] correct to two decimal place is,
\[\begin{align}
  & \sqrt{\dfrac{{{6}^{2}}}{{{\left( 2.23 \right)}^{2}}}} \\
 & \Rightarrow \dfrac{6}{2.23} \\
 & \Rightarrow 2.69 \\
\end{align}\]


Note: Students have to know the step of the long division method otherwise we cannot solve this problem. By this method students can calculate any non-perfect squares. Some students try to divide the given number first and then compute its square root, but that is simply going to complicate the whole process. So, it is better to find the square root of the numerator and denominator separately and then divide them.