Suppose m and n are whole numbers.  We will study the solution of the equation 

yⁿ = x^m

when one and hence both numbers x and y are non-zero.

Exercise: What can be said in the case where n and m are nonzero integers?

The sign function

Each real number q is positive, zero or negative.  We compute and hence define sign(q)  as follows.

So sign (5) = +1 and sign(-3) = -1 and sign(0) = 0.

Real Number Multiplication Revisited.

Multiple Distance to Origin &  Multiple Signs

Now  the product of a pair of real numbers a and b can be computed as follows.

ab  = [sign(a)sign(b)]  |a|*|b|

By mathematical induction we can show    

 t^k = [sign(t)]^k  |t|^k   and   

for all whole numbers k. For t non-zero, We can also show  sign(t)^k  = 1 when k = 2s is even for all real numbers t.  for all real t, we can show  sign(t)^k = sign(t) when k = 2s+ 1 is odd. 

Examples:  

• sign(5)⁴ = 1,  
• sign(-3)² = 1;   
• sign(5)³ = sign(5) and  
• sign(-2)⁷ = sign(-2)

Sign Analysis of the equation yⁿ = x^m

Applying the sign function to both sides of the equation   yⁿ = x^m forces 

 |y|ⁿ =  |x|^m

Applying the sign function to both sides of the equation   yⁿ = x^m gives 

sign(y)ⁿ |y|ⁿ = sign(x)^m |x|^m

So  sign(y)ⁿ = sign(x)^m 

Now y = sign(y)|y|.   The equation 

 |y|ⁿ =  |x|^m

implies    

n ln |y| =  ln |y|ⁿ =  ln  |x|^m = m ln |x|

Therefore n ln |y| =  = m ln |x| and     

Hence   

That says how to compute |y|.

Now we sign(y) from the equation 

sign(y)ⁿ = sign(x)^m 

• n-odd case:  If  n is odd,  sign(y) = sign(y)ⁿ = sign(x)^m for all real numbers x. Here  sign(x)^m will be 1 if m is even and sign(x) if m is odd.
• n-even case: If n is even, 1 = sign(y)ⁿ So the equation can only be satisfied when  1 = sign(x)^m  That is when x is positive with no restriction on m or when x is negative and m is even. Note: the equation 1 = sign(y)ⁿ allows y to be postive or negative.  So if y is a solution, so is -y, and when n is even, the equation  yⁿ = x^m  with x non zero has two solutions (a positive and negative) or no solutions. 

Conclusion I 

For  x < 0, the  equation  yⁿ = x^m has the positive solution  y = 0

---

When  x > 0 or m even the  equation  yⁿ = x^m has the positive solution  

y

= exp(

m  n

ln |x| )

and if n is even, the equation also has the negative solution  In the latter case, the positive solution is called the principal root.

y

= (-1) exp(

m  n

ln |x| )

---

For  x < 0 and m odd, the  equation  yⁿ = x^m for n even has no real solutions when n is even, and for n odd, it has the  solution  

y

= - exp(

m  n

ln |x| )

Conclusion II.

When m and n are both odd, and x is non-zero, the  equation  yⁿ = x^m has the solution  y = 0

y

= sign(x) exp(

m  n

ln |x| )

To see why, note the case x > 0 gives

y

= exp(

m  n

ln |x| )

while the case x < 0 gives

y

= - exp(

m  n

ln |x| )

in agreement with conclusion I.

Answer: When m and n are odd, and x is non-zero

When n is even,  and x > 0 

Remark:  For  x < 0 and m odd, the  equation  yⁿ = x^m for n even has no real solutions when n is even, but the equation

y²ⁿ = x^2m

has two solutions, the principal positive solution 

and its negative.

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