What is the square root of 90 simplified in radical form?

Question

Genesis Allen · Accepted Answer

#sqrt(90) = 3sqrt(10)#

To simplify #sqrt(90)#, the goal is to find numbers whose product gives the result of #90#, as well as collect pairs of numbers to form our simplified radical form.

In our case, we can begin in the following way:

#90 -> (30 * 3)#

#30 -> (10 * 3) # ... #*#... # 3#

#10 -> (5 * 2) # ...... # *#... # underbrace(3*3)_(pair) #

Since we don't have numbers we could further divide which yield a number other than #1#, we stop here and collect our numbers.

A pair of numbers counts as one number, namely the #3# itself.

Thus we can now write #sqrt(90) = 3sqrt(5*2) = 3sqrt(10)#

More examples:

(1) #sqrt(30)#

#30 -> (10 * 3)# #10 -> (5 * 2)# ... # * #... #3#

We cannot find any more divisible factors, and we certainly don't have a pair of numbers, so we stop here and call it not simplify-able. The one and only answer is #sqrt(30)#.

(2) #sqrt(20)#

#20 -> (10 * 2)# #10 -> (5) * underbrace(2 * 2)_(pair)#

We've found a pair, so we can simplify this one:

#sqrt(20) = 2sqrt(5)#

(3) #sqrt(56)#

#56 -> 8 * 7# #8 -> 4 * 2 * 7# #4 -> underbrace(2* 2)_(pair) * 2 * 7#

We proceed the same way and write #sqrt(56) = 2sqrt(2*7) = 2sqrt(14)#