Question

How do you simplify the square root of 200 – the square root of 32?

Answer 1

Hint: First of all, write the statement given in the above problem into mathematical expression. So, the square root of 200 is mathematically written as $\sqrt{200}$ and the square root of 32 is mathematically written as $\sqrt{32}$. Now, individually simplify $\sqrt{200}$ and $\sqrt{32}$ then subtract the second simplification of square root from the first one.

Complete step-by-step solution:
In the above problem, we are asked to simplify the subtraction of the square root of 32 from the square root of 200. So, we are first of all simplifying the square root of 200 and square root of 32.
Writing the square root of 200 we get,
$\sqrt{200}$
Now, we are going to write the prime factorization of 200.
$\Rightarrow 200=2\times 5\times 5\times 2\times 2$
Clubbing two same factors together in the above factorization and we get,
$\Rightarrow 200=2\times \underline{5\times 5}\times \underline{2\times 2}$
The underlined factors can be written as the square of one of the factors like we can write $5\times 5$ as ${{5}^{2}}$. Similarly, we can write $2\times 2$ as ${{2}^{2}}$.
$\Rightarrow 200=2\times {{5}^{2}}\times {{2}^{2}}$
Now, taking square root on both the sides of the above equation we get,
$\Rightarrow \sqrt{200}=\sqrt{2\times {{5}^{2}}\times {{2}^{2}}}$
In the above equation, we can take out the square terms written in the square root then the above equation will look like:
$\begin{align}
  & \sqrt{200}=\left( 5\times 2 \right)\sqrt{2} \\
 & \Rightarrow \sqrt{200}=10\sqrt{2} \\
\end{align}$
Hence, we have simplified $\sqrt{200}$ to $10\sqrt{2}$.
Similarly, we are going to simplify the square root of 32.
$\sqrt{32}$
Prime factorization of 32 is as follows:
$\Rightarrow 32=2\times 2\times 2\times 2\times 2$
Clubbing two same factors together in the above equation we get,
$\Rightarrow 32=\underline{2\times 2}\times \underline{2\times 2}\times 2$
Now, writing the underlined numbers as the square we get,
$\Rightarrow 32={{2}^{2}}\times {{2}^{2}}\times 2$
Taking square root on both the sides of the above equation we get,
$\Rightarrow \sqrt{32}=\sqrt{{{2}^{2}}\times {{2}^{2}}\times 2}$
Now, taking the square of the numbers out from the square root we get,
$\begin{align}
  & \sqrt{32}=\left( 2\times 2 \right)\sqrt{2} \\
 & \Rightarrow \sqrt{32}=4\sqrt{2} \\
\end{align}$
Hence, we have simplified the square root of 32 as $4\sqrt{2}$.
Subtracting the result of square root of 32 from the square root of 200 we get,
$10\sqrt{2}-4\sqrt{2}$
Taking $\sqrt{2}$ as common from the above expression we get,
$\begin{align}
  & \Rightarrow \sqrt{2}\left( 10-4 \right) \\
 & =\sqrt{2}\left( 6 \right) \\
 & =6\sqrt{2} \\
\end{align}$
Hence, the solution of the above problem is equal to $6\sqrt{2}$.

Note: The mistake that could be possible in the above problem is in writing the prime factorization of the numbers 200 and 32. Here, you might miss one or two factors of 200 and 32 so to rectify this problem, it’s better to multiply all the factors and then proceed.
For e.g. the factors which we have written for 200 is equal to:
$200=2\times 5\times 5\times 2\times 2$
To check whether we have missed any of the factors or not, we are going to multiply all the factors written on the R.H.S of the above equation.
$\Rightarrow 200=200$