MHB Square Root vs Cube Root

[mathdad]

I know that x^2 = 4 yields two answers: x = -2 or x = 2.

I also know that x^3 = 8 yields x = 2.

Question:

Why does the square root yield both a positive and negative answer whereas the cube root yields a positive answer?

 

[HallsofIvy]

The product of two positive or two negative numbers is positive. The product of three positive numbers is positive but the product of three negative numbers is negative.

By the way, if we allow complex numbers, then every number has n nth roots. If the original number is a positive real number, those roots lie on a polygon in the complex plane with n vertices, one of them the positive real root. If n is even, then there is another root on the negative real axis (for example, if n= 4 that is a square with one diagonal being the real axis) If n is odd the only real root is positive (if n= 3 we have an equilateral triangle and the real axis goes through the middle of the side between the two non-real roots.

Also, in an equation with all real coefficients, for every non-real root, its complex conjugate is also a root. That means there is always an even number of non-real roots. Since the principal root is real, if n is odd that is the only real root, if n is even, there is another real root.

 

[mathdad]

What about algebraic terms?

Say, sqrt{x^6}.

Can we say the answer is x^3 or -x^3 and x^3?

 

[MarkFL]

What about algebraic terms?

Say, sqrt{x^6}.

Can we say the answer is x^3 or -x^3 and x^3?

If $0\le x$, then we can call it $x^3$, otherwise, we call it $\left|x^3\right|$. :D

 

[HallsofIvy]

That strikes me as an odd way of phrasing it. If x\ge 0 then \sqrt{x^6} is x^3. If x< 0 then \sqrt{x^6}is -x^3. In either case, \sqrt{x^6} is |x^3|.

 

[mathdad]

If $0\le x$, then we can call it $x^3$, otherwise, we call it $\left|x^3\right|$. :D

If 0 is < or = x, then the answer is x^3. Otherwise, the answer must be the absolute value of x^3. Why is |x^3| the correct way to express the answer?

 

[MarkFL]

If 0 is < or = x, then the answer is x^3. Otherwise, the answer must be the absolute value of x^3. Why is |x^3| the correct way to express the answer?

Here is an article on the properties of square roots:

Square root

 

[mathdad]

Interesting.