Question

Find the square root of 1024. Hence find the value of $\sqrt {10.24} + \sqrt {0.1024} + \sqrt {10240000} $.

Answer 1

Hint: Here, we will be representing the number (whose square root is to be calculated) as the product of prime factors.

As we know that $1024 = {2^{10}}$
Taking square root on both sides on the above equation, we get
\[ \Rightarrow \sqrt {1024} = \sqrt {{{\left( 2 \right)}^{10}}} = {\left( 2 \right)^{\dfrac{{10}}{2}}} = {2^5} = 32\]
So, the square root of 1024 is 32.
Also, $\sqrt {10.24} + \sqrt {0.1024} + \sqrt {10240000} = \sqrt {\dfrac{{1024}}{{100}}} + \sqrt {\dfrac{{1024}}{{10000}}} + \sqrt {10240000} = \dfrac{{\sqrt {1024} }}{{\sqrt {100} }} + \dfrac{{\sqrt {1024} }}{{\sqrt {10000} }} + \left( {\sqrt {1024} } \right)\left( {\sqrt {10000} } \right)$
Since we already calculated \[\sqrt {1024} = 32\] and now using this, we get
$\sqrt {10.24} + \sqrt {0.1024} + \sqrt {10240000} = \dfrac{{32}}{{10}} + \dfrac{{32}}{{100}} + \left( {32} \right)\left( {100} \right) = 3.2 + 0.32 + 3200 = 3203.52$
Therefore the value of $\sqrt {10.24} + \sqrt {0.1024} + \sqrt {10240000} $ is 3203.52.

Note: The square root of division of two functions is equal to the division of square roots of these two functions i.e.,$\sqrt {\dfrac{{f\left( x \right)}}{{g\left( x \right)}}} = \dfrac{{\sqrt {f\left( x \right)} }}{{\sqrt {g\left( x \right)} }}$ and the square root of product of two functions is equal to the product of square root of these two functions i.e., $\sqrt {f\left( x \right)g\left( x \right)} = \left( {\sqrt {f\left( x \right)} } \right)\left( {\sqrt {g\left( x \right)} } \right)$.
The most common way to make mistakes in such problems is to inappropriately represent the decimal point.