Question

The value of $\sqrt {{{\left( {9999} \right)}^2} + 19999} $
1) 10,000
2) 9999
3) 100001
4) None of these

Answer 1

Hint: We will rewrite the given expression such that the expression inside the square-root is a perfect square. We can rewrite 19999 as $2\left( {9999} \right) + 1$. Then, we will have $\sqrt {{{\left( {9999} \right)}^2} + 2\left( {9999} \right)\left( 1 \right) + 1} $ . Also, 1 can be written as ${\left( 1 \right)^2}$. Thus, the expression $\sqrt {{{\left( {9999} \right)}^2} + 2\left( {9999} \right)\left( 1 \right) + 1} $ can be written as $\sqrt {{{\left( {9999} \right)}^2} + 2\left( {9999} \right)\left( 1 \right) + {{\left( 1 \right)}^2}} $which we can write using the formula, ${a^2} + 2ab + {b^2} = {\left( {a + b} \right)^2}$, as $\sqrt {{{\left( {9999 + 1} \right)}^2}} $

Complete step-by-step answer:
Since the expression has a square root, we can try making an expression of the whole square in square root, which cancels each other.
Here, we observe that 19999 is 1 more than twice of 9999.
We can rewrite 19999 as 19998+1 which is equivalent to $19998 + {\left( 1 \right)^2}$.
We are already given, ${\left( {9999} \right)^2}$ in the formula,
Hence, we can convert the given expression in the form, ${a^2} + 2ab + {b^2}$ which is equal to ${\left( {a + b} \right)^2}$.
We can observe that, 19998 is equals to $2\left( {9999} \right)\left( 1 \right)$
Therefore, we can rewrite the given expression as, $\sqrt {{{\left( {9999} \right)}^2} + 2\left( {9999} \right)\left( 1 \right) + 1} $.
We can convert the expression using the formula ${a^2} + 2ab + {b^2} = {\left( {a + b} \right)^2}$,
Here, $a = 9999$ and $b = 1$
$\sqrt {{{\left( {9999} \right)}^2} + 2\left( {9999} \right)\left( 1 \right) + 1} = \sqrt {{{\left( {9999 + 1} \right)}^2}} $
$\sqrt {{{\left( {9999} \right)}^2} + 2\left( {9999} \right)\left( 1 \right) + 1} = \sqrt {{{\left( {10,000} \right)}^2}} $
Also, \[\sqrt {{a^2}} = a\], therefore, $\sqrt {{{\left( {10,000} \right)}^2}} = 10,000$
Therefore, the value of $\sqrt {{{\left( {9999} \right)}^2} + 19999} $ is 10,000
Hence, option A is the correct answer.

Note: This question can alternatively be done by solving the square of 9999 and then adding to 19999. After that take the square root of the value to get the result.
$
  \sqrt {{{\left( {9999} \right)}^2} + 19999} = \sqrt {99980001 + 19999} \\
   = \sqrt {100000000} = 10000 \\
$