Squares and Roots V

$\frac { 1 }{ { m }^{ 2 } } =20$

$\frac { 1 }{ m } =\sqrt { 20 }$

If in the relation

$\frac { 1 }{ m } =\sqrt { n }$

$n=20$

$\frac { \sqrt { 20 } }{ 20 } =m$

Using the fact seen in the others problems (I does not say this but you can see comparing the solution of one problem with the first relation) of: "The square root of a number divided by this number is equals the square root of 1 divided by this number".

$\sqrt { \frac { 1 }{ 20 } } =m$

$m \times n = x$

$\sqrt { \frac { 1 }{ 20 } } \times 20\quad =\quad x$

$\frac { 1 }{ 20 } \times 400\quad =\quad { x }^{ 2 }$

$20={x}^{2}$

$\sqrt { 20 } =\quad x$

Factoring 20 we have:

$\sqrt { 20 } =2\times \sqrt { 5 } =\quad x\quad$

as from previous one,

m = 1 / (n)^0.5,

and given that 1 / m^(2) = 20<

m = 1 / (20)^0.5,

by comparing these two equations,

n = 20,

n x m = 20 x 1 / (20)^0.5 = 20^(0.5) = ( 4 x 5 )^(0.5)

= 4^(0.5) x 5^(0.5) = 2 x 5^(0.5)

therefore , the answer is 2 x 5^(0.5).....

thanks...

We know that:

$\frac{\sqrt{n}}{n} = m$

$\frac{1}{m} = \sqrt{n}$

According to question:

$\frac{1}{m^{2}} = 20$

Finding square roots of both sides:

$\frac{1}{m} = \sqrt{20}$ which can be compared to $\frac{1}{m} = \sqrt{n}$

$n = 20$

Now, using $\frac{\sqrt{n}}{n} = m$ we can find $m$ .

$\frac{\sqrt{20}}{20} = m$

$\frac{\sqrt{20}}{\sqrt{20} \times \sqrt{20}} = m$

$\frac{1}{\sqrt{20}} = m$

$m = \sqrt{\frac{1}{20}}$

So, $m = \sqrt{\frac{1}{20}}$ and $n = 20$ .

We need to find $m \times n$ . Let: $m \times n = x$

$m \times n = x$

$\sqrt{\frac{1}{20}} \times 20 = x$

Squaring both sides:

$x^{2} = \sqrt{\frac{1}{20}}^{2} \times 20^{2}$

$x^{2} = \frac{1}{20} \times 400$

$x^{2} = \frac{400}{20}$

$x^{2} = 20$

$x = \sqrt{20} = \sqrt{2^{2}\times 5} = 2 \times \sqrt{5}$

So, the answer is:

$m \times n = \sqrt{\frac{1}{20}} \times 20 = \sqrt{20} = \sqrt{2^{2}\times 5} = \boxed{2 \times \sqrt{5}}$