Question

How do you simplify $7\sqrt {600} $ ?

Answer 1

Hint: In this question, we have been asked to simplify a square root. Start by finding the prime factors of the number. Make the pairs of the prime factors, if possible. Those numbers, whose pairs are made, take one number out of each pair and write their product out of the square root. If any number is left, whose pairs cannot be made, multiply them and write their product in the square root.

Complete step-by-step solution:
We are given a square root and we have to simplify it.
Let us find its prime factors. Note that we will find the prime factors of only the number inside the square root.
Using prime factorisation method,
$\begin{array}{*{20}{c}}
  {2\left| \!{\underline {\,
  {600} \,}} \right. } \\
  {2\left| \!{\underline {\,
  {300} \,}} \right. } \\
  {2\left| \!{\underline {\,
  {150} \,}} \right. } \\
  {3\left| \!{\underline {\,
  {75} \,}} \right. } \\
  {5\left| \!{\underline {\,
  {25} \,}} \right. } \\
  {5\left| \!{\underline {\,
  5 \,}} \right. } \\
  {{\text{ }}1}
\end{array}$
Now, we can write $600 = 2 \times 2 \times 2 \times 3 \times 5 \times 5$.
Next step involves forming pairs of the prime factors.
$ \Rightarrow 600 = \left( {2 \times 2} \right) \times 2 \times 3 \times \left( {5 \times 5} \right)$
Now, those numbers whose pairs are available, will be written outside the square root and those numbers whose pairs cannot be made, will be written inside the square root.
Since the pairs of $2$ and $5$ are available, we will find their product and we will write the product outside the square root. And since the pairs of $2$ and $3$ are not available, we will find their product and we will write the product inside the square root.
Therefore, $7\sqrt {600} = 7 \times 2 \times 5\sqrt {2 \times 3} $
Simplifying it to find the answer
$ \Rightarrow 7\sqrt {600} = 70\sqrt 6 $

$70\sqrt6$ is the required answer.

Note: Instead of prime factorisation, we can also write $600 = 6 \times 100 = 6 \times {10^2}$. Then, we will simply have to write $10$ outside the square root and $6$ inside the square root. Though this method is applicable everywhere, it is easy only when you can differentiate the complete square from the given number. So, prefer prime factorisation to be on the safer side.