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Suppose m and n are whole numbers. We will study the solution of the
equation
yn = xm
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.
| sign(q) = |
{ |
+1 |
if q > 0 |
| 0 |
if q = 0 |
| -1 |
if q < 0 |
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
tk = [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 = xm
Applying the sign function to both sides of the equation yn
= xm forces
|y|n = |x|m
Applying the sign function to both sides of the equation yn
= xm 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 = xm with x non zero has two
solutions (a positive and negative) or no solutions.
Conclusion I
For x < 0, the equation yn = xm
has the positive solution y = 0
When x > 0 or m even the equation yn = xm
has the positive solution
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 = xm
for n even has no real solutions when n is even, and for n odd, it has
the solution
Conclusion II.
When m and n are both odd, and x is non-zero, the equation yn
= xm has the solution y = 0
| y |
= sign(x) exp( |
m
n |
|
ln |x| ) |
To see why, note the case x > 0 gives
while the case x < 0 gives
in agreement with conclusion I.
|
What is xb when b = |
m
n |
? |
|
Answer: When m and n are odd, and x is non-zero
| xb |
= sign(x) exp( |
m
n |
|
ln |x| ) |
When n is even, and x > 0
Remark: For x < 0 and m odd, the equation yn
= xm for n even has no real solutions when n is even, but the
equation
y2n = x2m
has two solutions, the principal positive solution
and its negative.
| y |
= (-1)exp( |
m
n |
|
ln |x| ) |
Animated Example
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