MathChat

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, etc.
 And … if you want to learn why this “hack” works, see my explanation at the end of the blog.

This “hack” lets you mentally do problems like the following three. That means you can do these problems in your head rather than on paper.

     a)  12 / √3 

     b)  10 / √2

     c)  22 / √5

Here are three terms I’ll use in explaining this “hack.”

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

     12  is  the dividend,

     3  is  the number under the radical,

     √3  is  the radical.

The “Hack,” Used for  12 / √3:

1.  Divide the dividend by the number under the radical.
   In this case, 12 / 3  =  4.
2. Take the answer, 4, and multiply it by the radical.
   4 x √3  =  4√3
   
   
3. Shake your head in amazement because that, right there, is the ANSWER!

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 get, 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 dividend by number under the radical.
   In this case,  22 divided by 5 = 22/5  (Yep, sometimes you wind up with a fraction or a decimal; that’s why I’m giving an example like this.)
2. Take the answer you get, 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:  22 √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

– – – – – – – – – – – – – – – – – –

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 “HACK” WORKS:

It works because we rationalize the denominator of a fraction whenever the denominator contains a radical. Here’s the “hack” 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 “hack” 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. I hope you are not offended.