Can I simplify square roots by using negatives?

Question

I was learning to simplify a square root by factoring within the radical in order to organize it more effectively,  such as sqrt(-52) = sqrt(-1)*sqrt(13)*sqrt(4) = 2(sqrt13)i ... however,  I was told it was incorrect to factor a positive number with two negatives, such as sqrt(16) = sqrt(-4)*sqrt(-4), as this will produce a nonsensical answer.  However,  I find it makes sense! 

 

sqrt(16) = sqrt(-4)*sqrt(-4) = 2i*2i = 4i^2 = -4... and since -4^2 = 16, it is not nonsense,  but just the opposite of the the principle root.

 

Was my tutor erroneous,  or am i?

Answer 1

It's PRINCIPAL root, not Principle root.

 

Your tutor was 100% correct.

 

Michael J's answer is NOT CORRECT; the principal square root of 16 is exactly 4 and nothing else.

 

It's a fundamental principle that the graph of the square root function is single-valued and that the domain is all Real non-negative numbers, and the Range is also all real non-negative numbers.

 

When learning about Imaginary numbers, all textbooks contain the warning similar to what your personal tutor gave you.

 

 

Answer 2

√(-4)√(-4) = (2i)(2i) = 4i² = -4

 

√(16) = -4

Since squaring a negative number gives you a positive. (-4)(-4) = 16