# How do you extract square roots in scientific notation?

Apr 2, 2016

See process and examples below.

#### Explanation:

In scientific notation numbers are written in the form $x \times {10}^{n}$, where $n$ is an integer and $x$ is in limits $\left[1 , 10\right)$ i.e. $1 \le x < 10$.

Examples$: -$

$53246.6$ is written as $5.32466 \times {10}^{4}$
$46870000$ is written as $4.687 \times {10}^{7}$
$0.0007925$ is written as $7.925 \times {10}^{-} 4$
$0.0000213$ is writen as $2.13 \times {10}^{-} 5$

To extract square root of such numbers

(a) if n is even just take the square root of $x$ and ${10}^{n}$ and multiply them; and

(b) if $n$ is odd, mutiply $x$ by $10$ and reduce $n$ by $1$ to make it even and then take square root of each and multiply them.

Hence

$\sqrt{5.32466 \times {10}^{4}} = \sqrt{5.32466} \times \sqrt{{10}^{4}} = 2.3075 \times {10}^{2}$
$\sqrt{4.687 \times {10}^{7}} = \sqrt{46.87} \times \sqrt{{10}^{6}} = 6.846 \times {10}^{3}$
$\sqrt{7.925 \times {10}^{-} 4} = \sqrt{7.925} \times \sqrt{{10}^{- 4}} = 2.815 \times {10}^{-} 2$
$\sqrt{2.13 \times {10}^{-} 5} = \sqrt{21.3} \times \sqrt{{10}^{- 6}} = 4.625 \times {10}^{-} 3$