Question

Solve the following questions and choose the correct option:-
1. \[\sqrt{99}\times \sqrt{396}\]
2. \[\sqrt{0.0324}\]

A. 192 ; 0.18
B. 192 ; 0.12
C. 198 ; 0.18
D. 198 ; 0.12

Answer 1

Hint: In these types of questions we are going to factorize the given numbers to primary numbers or small counterparts to make the question easy for us to solve. Meanwhile, we are looking for pairs of numbers, so that we can take them out of the square root and get rid of dealing with the irrational numbers or the square root. As the square root function is the inverse of the square function that is why we need a product pair of the same numbers in the square root function to end up with an integer as the answer.

Complete step by step answer:
1. Now in this question \[\sqrt{99}\times \sqrt{396}\] we are going to break these large numbers to small numbers by taking their L.C.M.
As their powers are the same and are square root so we will just make their power common and form a product of small numbers under a single power. Now as the power is square root so for every pair of a number only one of them comes out of the square root and gets the power of one and the other disappears from the equation, which is the role of the square root function.
Form as many prime number factors as you can to simplify the question to reduce the chances of mistake and save time while solving it.
L.C.M. of 99 is,
\[\begin{matrix}
   3 \\
   \begin{matrix}
   \begin{matrix}
   3 \\
   11 \\
\end{matrix} \\
   {} \\
\end{matrix} \\
\end{matrix}\left| \!{\underline {\,
  \begin{align}
  & 99 \\
 & 33 \\
 & 11 \\
 & 1 \\
\end{align} \,}} \right. \]

L.C.M. of 396 is,
\[\begin{matrix}
   2 \\
   \begin{matrix}
   \begin{matrix}
   2 \\
   3 \\
\end{matrix} \\
   \begin{matrix}
   \begin{matrix}
   3 \\
   11 \\
\end{matrix} \\
   {} \\
\end{matrix} \\
\end{matrix} \\
\end{matrix}\left| \!{\underline {\,
  \begin{align}
  & 396 \\
 & 198 \\
 & 99 \\
 & 33 \\
 & 11 \\
 & 1 \\
\end{align} \,}} \right. \]

Now by looking at the L.C.M. of 396 we can write 396 as \[4\times 99\] and now there is one pair of 99 and one pair of 2 inside the square root, and the question becomes \[\sqrt{99}\times \sqrt{4\times 99}=\sqrt{99}\times \sqrt{2\times 2\times 99}\]\[=\sqrt{2\times 2}\times \sqrt{99\times 99}\]
Now out of the each pair only one number will come out and hence the answer will be \[2\times 99=198\].


2. In this question we are going to find the answer using the same method used in the first part but here first we need to free our question from the decimal to simplify it.
Now as there are four digits after the decimal and to remove the decimal we are going to divide the whole number with \[{{10}^{4}}=10000\], one thing to keep in mind is that the number \[{{10}^{4}}\] will stay inside the square root. Now the real question is \[\sqrt{\dfrac{324}{{{10}^{4}}}}=\sqrt{\dfrac{324}{10000}}=\dfrac{\sqrt{324}}{\sqrt{10000}}\]
The L.C.M. of 324 is,

\[\begin{matrix}
   2 \\
   \begin{matrix}
   \begin{matrix}
   2 \\
   3 \\
\end{matrix} \\
   \begin{matrix}
   \begin{matrix}
   3 \\
   3 \\
\end{matrix} \\
   \begin{matrix}
   3 \\
   {} \\
\end{matrix} \\
\end{matrix} \\
\end{matrix} \\
\end{matrix}\left| \!{\underline {\,
  \begin{align}
  & 324 \\
 & 162 \\
 & 81 \\
 & 27 \\
 & 9 \\
 & 3 \\
 & 1 \\
\end{align} \,}} \right. \]


As we can tell by looking at the L.C.M. that there is one pair of 2 and two pairs of 3. So the numbers that will come out of the square root are \[2\times 3\times 3=18\].
Now 324 is the square of 18 if someone has memorized it but if not then we can go for L.C.M. again and find out the answer by pairing and eliminating. For \[{{10}^{4}}\] to get out from square root and achieve power of 1 it will form two pairs of \[(10\times 10)\times (10\times 10)\] or one pair of \[(100\times 100)\] and will come out as 100.
Hence the final answer becomes \[\dfrac{18}{100}=0.18\]

So the correct option is, C.
Answer C. \[198;0.18\]

Note:
 You can learn the values of squares of numbers and also the square root to save time while solving questions in the exam. Sometimes you may get a difficult question asking you the square root of prime numbers and you may require to learn them thoroughly.
Look out the number of digits after the decimal, and do not count the zeroes at the end, and carefully divide the number by the 10 raised to the power of the number of digits in the numerator after the decimal. After dividing the number with 10 raised to the power of numerator digits, remove the decimal and you will get an integer.
When there is a cube root and we want to solve it then go for a group of three same numbers in the product to get rid of the cube root and end up with an integer as the answer. Similarly, for a double square root or raise to the power of one-quarter (0.75) go for four same numbers in product and so on.