root question

**portia** · September 27th 2009, 09:58 AM

Can anyone explain the following:

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

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

thanks in advance

How on earth is A) true?

**MacstersUndead** · September 27th 2009, 10:03 AM

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?

**e^(i*pi)** · September 27th 2009, 10:14 AM

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

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

**Grandad** · September 27th 2009, 10:24 AM

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

thanks in advance

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 $\sqrt2$ :

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

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

$=\sqrt2:1$

Grandad

**portia** · September 27th 2009, 10:25 AM

Thanks, I understand it now. After substracting the exponents we're left with 2^(-1/2) which gives 1 over the square root of 2.

My maths skills are VERY rusty.

What about the ration question?

edit: sorry, I only saw the first reply.

**portia** · September 27th 2009, 11:36 AM

Thanks guys, I get it now

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