Question

$\text{Find the value of}(\sqrt{49}+\sqrt{0.49}+\sqrt{0.0049})\times \sqrt{10000}.$

Answer 1

$\text{The given expression can be written as}$

$(\sqrt{49}+\sqrt{\frac{49}{100}}+\sqrt{\frac{49}{10000}})\times \sqrt{10000}$

$\text{Prime factorisation of 10000 is:}$
$\begin{array}{cc}2& 10000\\ 2& 5000\\ 2& 2500\\ 2& 1250\\ 5& 625\\ 5& 125\\ 5& 25\\ 5& 5\\ & 1\end{array}$

$10000=2\times 2\times 2\times 2\times 5\times 5\times 5\times 510000=10\times 10\times 10\times 10\sqrt{10000}=2\times 2\times 5\times 5=10\times 10$

$(\sqrt{49}+\sqrt{\frac{49}{100}}+\sqrt{\frac{49}{10000}})\times \sqrt{10000}$

$=(\sqrt{7\times 7}+\sqrt{\frac{7\times 7}{10\times 10}}+\sqrt{\frac{7\times 7}{10\times 10\times 10\times 10}})\times \sqrt{10\times 10\times 10\times 10}$

$=(7+\frac{7}{10}+\frac{7}{10\times 10})\times 10\times 10$

$=(7+0.7+0.07)\times 100$

$=7.77\times 100=777$