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
$$\Rightarrow \sqrt{1024}=\sqrt{{\left(2\right)}^{10}}={\left(2\right)}^{{\displaystyle \frac{10}{2}}}=2^5=32$$
So, the square root of 1024 is 32.
Also, $\sqrt{10.24}+\sqrt{0.1024}+\sqrt{10240000}=\sqrt{{\displaystyle \frac{1024}{100}}}+\sqrt{{\displaystyle \frac{1024}{10000}}}+\sqrt{10240000}={\displaystyle \frac{\sqrt{1024}}{\sqrt{100}}}+{\displaystyle \frac{\sqrt{1024}}{\sqrt{10000}}}+\left(\sqrt{1024}\right)\left(\sqrt{10000}\right)$
Since we already calculated $$\sqrt{1024}=32$$ and now using this, we get
$\sqrt{10.24}+\sqrt{0.1024}+\sqrt{10240000}={\displaystyle \frac{32}{10}}+{\displaystyle \frac{32}{100}}+\left(32\right)\left(100\right)=3.2+0.32+3200=3203.52$
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.,$\sqrt{{\displaystyle \frac{f\left(x\right)}{g\left(x\right)}}}={\displaystyle \frac{\sqrt{f\left(x\right)}}{\sqrt{g\left(x\right)}}}$ 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.