Question

What is the simplest radical form of $ 2\sqrt {112} $ ?

Answer 1

Hint: In order to find the simplest radical form of the given number first we need to solve the square root by expanding the radicand in terms of its prime factors and then taking out the common values whose square roots are known and simply multiplying it with $ 2 $ .

Complete step-by-step answer:
We are given with the number $ 2\sqrt {112} $ .
Solving the radicand that is $ \sqrt {112} $ by expanding it in terms of its prime factors and we get that the prime factors of $ 112 $ is $ 2 \times 2 \times 2 \times 2 \times 7 $ .
Putting the factors inside the Square root:
 $ \sqrt {112} = \sqrt {2 \times 2 \times 2 \times 2 \times 7} $ .
Since, we know that two similar numbers present inside the root can be taken out as a single unit, what we call its square root. For ex: $ \sqrt {x \times x} = x $ .
Similarly, we know that $ \sqrt 4 = \sqrt {2 \times 2} = 2 $ .
As we can see that there are $ 2 \times 2 \times 2 \times 2 $ , so we can take out two 2’s from the root and we are left with $ 7 $ inside that is: $ \sqrt {2 \times 2 \times 2 \times 2 \times 7} = 2 \times 2\sqrt 7 = 4\sqrt 7 $ .
Substituting this value in the original question that is $ 2\sqrt {112} $ and we get:
 $ 2\sqrt {112} = 2\left( {4\sqrt 7 } \right) $
Multiplying the terms outside roots, that is $ 2 $ and $ 4 $ , we obtained:
 $
  2\sqrt {112} = 2\left( {4\sqrt 7 } \right) \\
  2\sqrt {112} = 2 \times 4\sqrt 7 \\
  2\sqrt {112} = 8\sqrt 7 \;
  $
We can further simplify $ \sqrt 7 $ , or can leave it here only.
Therefore, the simplest radical form of $ 2\sqrt {112} $ is $ 8\sqrt 7 $ .
So, the correct answer is “ $ 8\sqrt 7 $ ”.

Note: We can leave the solution in roots only or can simplify the roots and then multiply the terms outside roots.
If the radicand cannot be further expanded or it’s a prime number then leave it as it is.
Always cross check the answer once.
Radicand is the term or value that is present inside the roots.