Question

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

Answer 1

$\begin{array}{cc}2& 1024\\ 2& 512\\ 2& 256\\ 2& 128\\ 2& 64\\ 2& 32\\ 2& 16\\ 2& 8\\ 2& 4\\ 2& 2\\ & 1\end{array}$

From prime factorisation,

$\sqrt{1024}=\sqrt{(2\times 2)\times (2\times 2)\times (2\times 2)\times (2\times 2)\times (2\times 2)}$

$\Rightarrow \sqrt{1024}=2\times 2\times 2\times 2\times 2$

$\Rightarrow \sqrt{1024}=32$

Now, $\sqrt{10.24}$

$=\sqrt{\frac{1024}{100}}$ [Since, $10.24$ contain two number of decimals.]

$=\frac{32}{10}$

$=3.2$ [Since, Shifted the decimal point one step towards left]

$\Rightarrow \sqrt{10.24}=3.2$---(1)

$\sqrt{0.1024}$

$=\sqrt{\frac{1024}{10000}}$ [Since, $0.1024$ contain four number of decimals]

$=\frac{32}{100}$

$=0.32$ [Since, Shifted the decimal point two steps towards left]

$\sqrt{0.1024}=0.32$---(2)

$\sqrt{10240000}$

$=\sqrt{1024\times 10000}$

$=\sqrt{1024}\times \sqrt{10000}$

$=32\times 100$

$=3200$

$\Rightarrow \sqrt{10240000}=3200$ ---(3)

Now, from (1), (2) and (3)

$\sqrt{10.24}+\sqrt{0.1024}+\sqrt{10240000}$

$=3.2+0.32+3200$

$=3203.52$

$\therefore \sqrt{10.24}+\sqrt{0.1024}+\sqrt{10240000}=3203.52$