1. ## root question

Can anyone explain the following:

A)
$\frac{\sqrt{2}}{2}=\frac{1}{\sqrt{2}}$

and the ratio/proportion:
B)
$1$ : $\frac{1}{\sqrt{2}}$ = $\sqrt{2}$ : $1$

How on earth is A) true?

2. A) You can rewrite this equation with exponents

Consider that the square root of 2 = 2^(1/2)
now with the exponent law of division, can you finish?

3. $\frac{\sqrt2}{\sqrt2} = 1$

And as we know multiplying by 1 does not affect the value of a number.

$\frac{1}{\sqrt2} \times \frac{\sqrt2}{\sqrt2} = \frac{\sqrt2}{2}$

You'll come across this more when it comes to Rationalising the Denominator

4. Hello portia
Originally Posted by portia
Can anyone explain the following:

A)
$\frac{\sqrt{2}}{2}=\frac{1}{\sqrt{2}}$

and the ratio/proportion:
B)
$1$ : $\frac{1}{\sqrt{2}}$ = $\sqrt{2}$ : $1$

How on earth is A) true?
The only things you really need to understand here are:

• $\sqrt2\times\sqrt2 = 2$ (because that's what a square root is)

• how to 'cancel' a fraction

Then it works like this:

$\frac{\sqrt2}{\color{red}2}=\frac{\sqrt2}{\color{r ed}\sqrt2\times\sqrt2}$

... and 'cancel' in the usual way:

$\frac{\sqrt2}{2}=\frac{\color{blue}\sqrt2}{\color{ blue}\sqrt2\color{black}\times\sqrt2}=\frac{1}{\sq rt2}$

For B, you also need to understand that ratios work exactly like fractions: you can 'cancel' (i.e. divide each number in the ratio by the same thing) or you can 'un-cancel' (if there is such a word) by multiplying each number in the ratio by the same thing.

So starting with $1:\frac{1}{\sqrt2}$, multiply each number by $\sqrt2$:

$1:\frac{1}{\sqrt2}=1\color{red}\times\sqrt2\color{ black}:\frac{1}{\sqrt2}\color{red}\times \sqrt2$

$= \sqrt2:\frac{\sqrt2}{\sqrt2}$

$=\sqrt2:1$