## Kiss those Math Headaches GOODBYE!

### How to Divide ANY Number by a Radical — Fast!

Here’s a super-quick shortcut for  DIVIDING ANY NUMBER by a RADICAL.

Note: I’m using this symbol () to mean square root.
So √5 means the square root of 5;  √b means the square root of b.
And … if you want to learn why this shortcut works, see my explanation at the end of the blog.

This shortcut lets you mentally do problems like the following three problems. That means you can do such problems in your mind rather than having to work them out on paper.

a)  12 / √3

b)  10 / √2

c)  22 / √5

Here are three terms I’ll use in explaining this shortcut.

In a problem like 12 divided by √3, which I am writing as:  12 / √3,

12  is  the dividend,

3  is  the number under the radical,

## The Shortcut, Used for  12 / √3:

1.  Divide the dividend by the number under the radical.
In this case, 12 / 3 = 4.
2. Take the answer you got, 4, and multiply it by the radical.
4 x √3  =  4√3

## Another Example:  10 / √2

1.  Divide the dividend by the number under the radical.
In this case:   10 / 2  =  5
2. Take the answer you got, 5, and multiply it by the radical.
5 x √2  =  5√2.  (Don’t forget to shake head in amazement!)

## Third Example:  22 / √5

1.  Divide the dividend by the number under the radical.
In this case,  22 divided by 5 = 22/5  (Yup, sometimes you wind up with a fraction or with a decimal; that’s why I’m giving an example like this.)
2. Take the answer you got, 22/5, and multiply it by the radical.
22/5 x √5 =  22/5 √5.  [Note: the √5 is in the numerator, not
in the denominator. To make the location of this √5 clear, it’s best
to write the answer:  2√5 / 5].

## NOW TRY YOUR HAND by doing these PRACTICE PROBLEMS:

a)   18 / √3

b)   16 / √2

c)   30 / √5

d)   10 / √3

e)   12 / √5

– – – – – – – – – – – – – – – – – –

a)   18 / √3  = 6√3

b)   16 / √2  = 8√2

c)   30 / √5  = 6√5

d)   10 / √3  = 10√3/3

e)   12 / √5  = 12√5/5

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## WHY THE SHORTCUT WORKS:

The shortcut works because we rationalize the denominator of a fraction whenever the denominator contains a radical. Here’s the shortcut in general terms, with:

a  =  the dividend,

b  =  the number under the radical,

a / √b

=   a
√b

=   a     √b    =   a √b
√b   √b            b

Notice: we started with:  a / √b.

And keeping things equal, we ended up with  a √b / b.

This shows that the shortcut works in general. So it works in all specific cases as well!

– – – – – – – – – – – – – – – – – –

Final note: the number under the radical is called the radicand. But that term is so close to the term radical that I thought it would be less confusing if I just called this the number under the radical.