To divide powers with the same nonzero base, we subtract the exponents.
x=7 and y=4, see solution.
Practice makes perfect
Our exercise involves the expressions shown below.
a^x/a^y=a^3 and a^x/a^(3y)=a^(- 5)
Therefore, it will be useful to start by reviewing the property for dividing powers with same base.
To divide powers with the same base, subtract the exponents.
a^m/a^n = a^(m-n)
This is valid if a≠0 and m and n are rational numbers.
If we use this property we can rewrite the fractions on the left-hand side of the equations as a single power with base a. This will allow us to compare the exponents directly and set up a system of equations to solve for x and y. Let's try with a^xa^y=a^3 first.
With this second equation we can set up a linear system of two equations and two variables.
x-y=3 & (I) x-3y=-5 & (II)
Notice that the coefficient for x is the same in both equations. Therefore, we can use the Elimination Method to solve it. To do this, we can subtract Equation (II) from Equation (I).
