Question

Simplify: $ 3\sqrt {45} - \sqrt {125} + \sqrt {200} - \sqrt {50} $

Answer 1

Hint: For this type of problem we should first factor the radicand of each radical term and after that adding and subtracting multiple of radical terms having the same radicand formed after simplification and hence required value of given problem.

Complete step-by-step answer:
Given, $ 3\sqrt {45} - \sqrt {125} + \sqrt {200} - \sqrt {50} $
To simplify it. We will first form the prime factor of each radical term of the radical term given in the equation.
Prime factors of term $ \sqrt {45} \,\,\,is\,\,given\,\,as\,\,\sqrt {3 \times 3 \times 5} $
Or we can write $ \sqrt {45} $ as $ 3\sqrt 5 $ ……………(i)
Prime factors of term $ \sqrt {125} $ is given as $ \sqrt {5 \times 5 \times 5} $
Or we can write $ \sqrt {125} $ as $ 5\sqrt 5 $ ……….(ii)
Prime factors of term $ \sqrt {200} $ is given as $ \sqrt {2 \times 2 \times 2 \times 5 \times 5} $
Or we can write $ \sqrt {200} $ as $ 10\sqrt 2 $ ……..(iii)
Prime factors of term $ \sqrt {50} $ is given as $ \sqrt {5 \times 5 \times 2} $
Or we can write $ \sqrt {50} $ as $ 5\sqrt 2 $ ………….(iv)
Using all these values obtained in equation (i) , (ii), (iii) and (iv) in the given equation. We have,
 $ 3\left( {3\sqrt 5 } \right) - \left( {5\sqrt 5 } \right) + \left( {10\sqrt 2 } \right) - \left( {5\sqrt 2 } \right) $
 $
   \Rightarrow 9\sqrt 5 - 5\sqrt 5 + 10\sqrt 2 - 5\sqrt 2 \\
   \Rightarrow \left( {9 - 5} \right)\sqrt 5 + \left( {10 - 5} \right)\sqrt 2 \\
   \Rightarrow 4\sqrt 5 + 5\sqrt 2 \;
  $
Hence, from above we see that the required value of $ 3\sqrt {45} - \sqrt {125} + \sqrt {200} - \sqrt {50} $ is $ 4\sqrt 5 + 5\sqrt 2 $ .

So, the correct answer is “ $ 4\sqrt 5 + 5\sqrt 2 $ ”.

Note: To simplify radical terms we have to make radicand same if they are not same and this can be done only by making prime factors of radicand of each radical terms and then adding and subtracting those radical terms having same radicands and so solution of given problem.