For what do we use the term real root? For what do we use the term principal root?
See solution.
Practice makes perfect
Consider the polynomial equation x^n=b. The Fundamental Theorem of Algebra tells us that this polynomial equation has n roots, which can be either real or imaginary. When restricting ourselves to just the real ones, we refer to a number a, such that a^n=b, as the real nth root of b. There are two different cases.
Solving a^n=b When n Is Odd
If n is odd, there is just one real nth root of b, which we denote as sqrt(b).
Equation
Real Solutions
nth Root
x^3=8
x=2
sqrt(8)=2
x^3=- 27
x=- 3
sqrt(-27)=- 3
Solving a^n=b When n Is Even
If n is even, there are two possibilities, depending on if b is positive or negative. If b is positive there are two possible real nth roots. The positive root is know as the principal root.
Equation
Real Solutions
nth Root
Root Type
x^4=16
x=2
sqrt(16)= 2
Principal Root
x=- 2
- sqrt(16)= -2
Negative Root
On the other hand, if b is negative, there are no real solutions. All the nth roots are imaginary in that case.
Conclusions
A solution to the equation x^n=b is called the nth root of b.
We use the term real nth root to differ between real and complex possible roots.
We use the term principal root to refer to the positive real root in cases when we can have a positive and a negative real root.
