Algebra's Laws of Powers and Roots: Handle With Care

• Basics

Recently, for entertainment, I tried to solve a puzzling algebra problem featured on YouTube [1]. In due course I learned that algebra’s $$(a^x)^y=a^{xy}\qquad\qquad\qquad\qquad\qquad(1)$$

Law of Powers identity is not always valid (not always true) if variable a is real and exponents x and y are complex-valued.

The fact that Eq. (1) can’t reliably be used with complex x and y exponents surprised me. And then I thought, “Humm, …what other of algebra’s identities may also be invalid when the exponents are complex?” This blog provides a table to answers that question.

I’ve examined several algebra identities and tabulated under what conditions they are invalid for two scenarios. In Scenario# 1 the x and y exponents are real-valued, and in Scenario# 2 the x and y exponents are complex-valued whose real and imaginary parts are real numbers. In the following table the mantissa scalar variables a and b are real-valued.

This blog’s appendix gives MATLAB command examples that verify the entries in the above table.

My Test Method

For Test Scenario# 1 I tested the validity of the above algebra identities using the following steps:

[1] Assigned real bipolar values to scalar variables a and b. Those values ranged from ‑10 to +10 not including zero.

[2] Assigned random real values to exponents x and y. Those real values had magnitudes in the range of zero to 20.

[3] Using the assigned a, b, x, and y variables, compute the numerical values of the left and right sides of an algebra identity using MATLAB. (When computing roots MATLAB computes the principle root.)

[4] Compute the difference (error) between the two computed numbers from Step# 3.

[5] Repeat Steps [1] through [4] thousands of times for various random numerical combinations of the x and y exponents.

For Test Scenario# 2 I performed the above Steps using random complex values for the x and y exponents. Those complex values had magnitudes in the range of zero to 20 and random angles in the range of zero to 2p radians.

Conclusion

The validity of algebra’s Laws of Powers and Roots identities, with real or complex exponents, depend on the polarities of the a and b mantissa scalars. I found no discernable pattern in which x and y values caused any given identity to be invalid.

My conclusion for this blog is: In your mathematical derivations, be cautious when using algebra’s Laws of Powers and Roots.

Acknowledgements

I thank Randy Yates and Dan Boschen for their advice regarding the content of this blog.

References

[1] “Breaking the rules of math π = 0. Where is the mistake?” Online:

https://www.youtube.com/watch?v=R476CTKUIr4&ab_channel=MindYourDecisions

Appendix

This appendix provides example MATLAB commands that verify the entries in the above table. The following Left and Right variables are the computed left and right sides of a given identity. A MATLAB-computed result where Left ≠ Right is proof that a given identity is “Not always valid.”

Identity# 1a:

  Real x and y:

   a = -2; x = 2; y = 2; Left = a^(x/y), Right = (a^x)^(1/y)

  Complex x and y:

   a = 2; x = -4 -9i; y = -5 + 6i; Left = a^(x/y), Right = (a^x)^(1/y)

   a = -2; x = -4 -9i; y = -5 + 6i; Left = a^(x/y), Right = (a^x)^(1/y)

Identity# 1b:

  Real x and y:

   a = -2; x = 1.1; y = 0.7; Left = a^(x/y), Right = (a^(1/y))^x

  Complex x and y:

   a = -2; x = 3 + 4i; y = -1 + 0i; Left = a^(x/y), Right = (a^(1/y))^x

Identity# 2:

  Real x and y:

   a = -2; b = -5; x = -2.2; Left = (a*b)^x, Right = (a^x)*(b^x)

  Complex x and y:

   a = -2; b = -5; x = 1 + 4i; Left = (a*b)^x, Right = (a^x)*(b^x)

Identity# 3:

  Real x and y:

   a = 2; b = -5; x = 1.3; y = 1.02; Left = (a^x)*(b^(x*y)), Right = (a*b^y)^x

   a = -2; b = -5; x = 1.3; y = 1.02; Left = (a^x)*(b^(x*y)), Right = (a*b^y)^x

  Complex x and y:

   a = 2; b = 5; x = 6 + 2i; y = 0 -7i; Left = (a^x)*(b^(x*y)), Right = (a*b^y)^x

   a = 2; b = -5; x = 6 + 2i; y = 0 -7i; Left = (a^x)*(b^(x*y)), Right = (a*b^y)^x

   a = -2; b = 5; x = 6 + 2i; y = 0 -7i; Left = (a^x)*(b^(x*y)), Right = (a*b^y)^x

   a = -2; b = -5; x = 6 + 2i; y = 0 -7i; Left = (a^x)*(b^(x*y)), Right = (a*b^y)^x

Identity# 4:

  Real x and y:

   a = 2; b = -5; x = 1.4; Left = ((a)/(b))^x, Right = ((a)^(x))/((b)^(x))

  Complex x and y:

   a = 2; b = -5; x = 1 + 2i; Left = ((a)/(b))^x, Right = ((a)^(x))/((b)^(x))

Identity# 5:

  Real x and y:

   a = -2; x = 1.1; y = 0.7; Left = (a^x)^y, Right = a^(x*y)

  Complex x and y:

   a = 2; x = 2 + 5i; y = 3 -12i; Left = (a^x)^y, Right = a^(x*y)

   a = -2; x = 2 + 5i; y = 3 -12i; Left = (a^x)^y, Right = a^(x*y)

Identity# 6:

  Real x and y:

   a = -2; b = -5; x = -0.8; Left = (a*b)^(1/x), Right = (a^(1/x))*(b^(1/x))

  Complex x and y:

   a = -2; b = -5; x = -1 - 3i; Left = (a*b)^(1/x), Right = (a^(1/x))*(b^(1/x))

Identity# 7:

  Real x and y:

   a = -2; x = 1.1; y = 0.7; Left = (a^(1/y))^(1/x), Right = a^(1/(x*y))

  Complex x and y:

   a = 2; x = -3 - 4i; y = 0 + 0i; Left = (a^(1/y))^(1/x), Right = a^(1/(x*y))

   a = -2; x = 10 + 8i; y = -1 + 0i; Left = (a^(1/y))^(1/x), Right = a^(1/(x*y))

Identity# 8:

  Real x and y:

   a = 2; b = -5; x = 1.4; Left = (a/b)^(1/x), Right = (a^(1/x))/(b^(1/x))

  Complex x and y:

   a = 2; b = -5; x = 2 + 1i; Left = (a/b)^(1/x), Right = (a^(1/x))/(b^(1/x))

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