This equation shows how to simplify division of exponents
| \( a \) | This is a symbol for any generic constant. It can hold any numerical value |
| \( x \) | This is a symbol for any generic variable. It can hold any value, whether that be an integer or a real number, or a complex number, or a matrix etc. |
| \( y \) | This is a secondary symbol for any generic variable. It can hold any value, whether that be an integer or a real number, or a complex number, or a matrix etc. |
We will show this to be the case intuitively by an example. For our equation:
\[\frac{\htmlClass{sdt-0000000121}{a}^{\htmlClass{sdt-0000000003}{x}}}{\htmlClass{sdt-0000000121}{a}^{\htmlClass{sdt-0000000017}{y}}} = \htmlClass{sdt-0000000121}{a}^{\htmlClass{sdt-0000000003}{x} - \htmlClass{sdt-0000000017}{y}}\]
Let us say:
We can now say that we are trying to find:
\[\frac{3^5}{3^2}\]
We can now use the definition of exponentiation:
\[\htmlClass{sdt-0000000003}{x}^{\htmlClass{sdt-0000000017}{y}} = \htmlClass{sdt-0000000097}{\prod}_{\htmlClass{sdt-0000000018}{i} = 1}^{\htmlClass{sdt-0000000017}{y}} \htmlClass{sdt-0000000003}{x}\]
And expand to get:
\[\frac{3^5}{3^2} = \frac{3 \cdot 3 \cdot 3 \cdot 3 \cdot 3}{3 \cdot 3}\]
Now two of the \(3\) can cancel out from both the numerator and denominator:
\[\frac{3^5}{3^2} = \frac{\cancel{3} \cdot \cancel{3} \cdot 3 \cdot 3 \cdot 3}{\cancel{3} \cdot \cancel{3}} = 27\]
By looking at this, you can see that cancelling out identical effect to subtracting the exponents:
\[\frac{3^5}{3^2} = 3^{5-2} = 3^3 = 27\]
Therefore:
\[\frac{\htmlClass{sdt-0000000121}{a}^{\htmlClass{sdt-0000000003}{x}}}{\htmlClass{sdt-0000000121}{a}^{\htmlClass{sdt-0000000017}{y}}} = \htmlClass{sdt-0000000121}{a}^{\htmlClass{sdt-0000000003}{x} - \htmlClass{sdt-0000000017}{y}}\]
as required.