Dear Peter Breuer, I couldn't understand your concept, please give detailed explanation according to my question or give some other example.

Thank you for your answer

take x as rotation operation by an arbitrary angle (theta) about z axis .

then x^2=x*x correspond to rotation about z axis by 2*theta angle.

x^3=x*x*x correspond to rotation about z axis by 3*theta angle

hence, x^2*x*3 correspond to rotation about z axis by 5*theta angle, whixh is equal to x^5=x*x*x*x*x.

Dear Kumar,

I will consider a geometrical interpretation assuming X is real

and the exponents are positive integers.

Let us discuss the general form:

What is the meaning of Xm ?

Xm=X×X...×X m-times.

It is the Cartesian product of m -intervals.

For X is a real number then

X can be considered as the length of an interval.

X²=X×X is the area of a square.

X³=X×X×X is the volume of a cube.

In general: For m > 3 Xm is the volume of a cuboid with the edge X

in dimension m.

Geometrically: We can say X³×X²:

(the cube of dimension 3 and edge X) times (the square of dimension 2 and edge X) is a cuboid X2+3 =X5 of a dimension 2 + 3 = 5 and edge X.

An interesting cuboid in dimension 4 ( X4 : called Tesseract)

See the attached graph.

Best regards

@Wulf,

Exponents of real numbers X n can be considered as expansions and

contractions according to X is greater or less than one.

Also, it can be considered as transformations Zn for Z is a complex number Z= rn eitheta as tackled by you, Gosh and Peter.

I like to add some geometry to the answer which is accepted by secondary school students where higher dimensions are attractive as in the case

of the TESSERACT.

All contributions are helpful to reach a unified answer :)

Best wishes

Lots of explanations given are fun but requires much more mathematical maturity to understand and digest... I would just write it the expanded way:

x^2 = x*x

x^3 = x*x*x

Therefore, x^2*x^3 = (x*x) * (x*x*x) = x^5 (assuming your students already know the exponential notation of course).

In the end, it's just a property of multiplication and all other examples are kind of multiplications in disguise but the disguise makes them more difficult to grasp!

Dear Issam kaddoura, Thank you for your nice and understanding reply.