For example, a8/a3=a5. Explain and show your reasoning to prove this property to be true for all values of a.

Question

The quotient of powers property states that to divide powers with the same base, subtract their exponents. For example, a8/a3=a5. Explain and show your reasoning to prove this property to be true for all values.

Answer 1

We are asked to investigate a⁸/a³=a⁵ 

 

We can expand these expressions to obtain 

 

a.a.a.a.a.a.a.a/a.a.a

 

Since a/a =1, we can rewrite these expressions as

 

a/a . a/a . a/a . a.a.a.a.a

 

= 1.1.1.a.a.a.a.a

 

and since 1.1=1, we now have

 

a.a.a.a.a, which we can rewrite as a⁵.

 

Since we can always expand an exponential expression, and we can always cancel identical numerators and denominators, we can always divide by subtracting exponents.