### 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,**

** √3 is the radical.**

**The Shortcut, Used for 12 / √3:**

- Divide
**the dividend**by t**he number under the radical**.

In this case,**12 / 3 = 4.** - Take the answer you got,
**4**, and multiply it by**the radical**.

**4 x √3 = 4****√3**

- Shake your head in
**amazement**because that, right there, is the ANSWER!

**Another Example: 10 / √2**

- Divide the dividend by the number under the radical.

In this case:**10 / 2 = 5** - Take the answer you got,
**5**, and multiply it by the radical.

**5 x √2 = 5**Don’t forget to shake head in amazement!)**√2. (**

**Third Example: 22 / √5**

- 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.)** - Take the answer you got,
**22/5**, and multiply it by the radical.

**22/5 x √5 = 22/5**[Note: the √5 is in the numerator, not**√5.**

in the denominator. To make the location of this √5 clear, it’s best

to write the answer: 2].**2****√5 / 5**

*NOW TRY YOUR HAND* *by doing*

*these PRACTICE PROBLEMS:*

*NOW TRY YOUR HAND*

*by doing*

*these PRACTICE PROBLEMS:*

a) ** 18 / √3 **

b) ** 16 / √2 **

c) ** 30 / √5 **

d) ** 10 / √3 **

e) ** 12 / √5**

– – – – – – – – – – – – – – – – – –

**ANSWERS:**

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**

– – – – – – – – – – – – – – – – – –

**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,**

** √b = 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.*