Expert: Sherry Wallin - 1/22/2009

Question
I just started my 2nd semester of my online school and I am already having trouble with this part of Algebra 2.  I need to know how to find the square and cube roots of a number.  Example: http://schools.connectionsacademy.com/content/media/95736-9202006-93401-AM-19819...  I need to find all real square roots of that number along with 121, but I already know that the sqrt of 121 is 11 and -11.  Also, I need to be able to find all real cube roots of http://schools.connectionsacademy.com/content/media/95736-9202006-95844-AM-16721...   I don't even know how to read that!! Please help me!?  I am like freaking out.

Thanks,
Kadie

Answer
Hi Kadie~
    You can not take the even root of any negative number. This means you cannot take the square (2nd) root or the 4th root or the 6th root etc. So the square root of (-1/25) DOES NOT EXIST. Why? Because there is no number that when you square it (multiply it by itself)will give you a negative number. Remember two negative numbers multiplied together give you a positive number and a negative and a positive multiplied together will give you a negative number and two positives multiplied together give you a positive so you see there is no number that can be multiplied by itself that is negative. Again this is true for all even roots. However not so with odd roots such as cube roots. The sign on the number you want to take the cube root of will be the sing of the cube root. Example: the cube root of 8 is 2 whereas the cube root of -8 is -2. So you want to know how to find the cube root of 125. You already know it will be a positive number. You are looking for a number times itself times itself that gives you 125. But why not prime factor the number and find out the answer that way? 125 = 5^3 which does tell you that the cube root of 125 is 5 but what if it was the cube root of 54? The prime factorization of 54 = 2*3^3 so the cube root of 54 is 3*cube root of 2.

Math Prof