## anonymous one year ago Friends... Please help me with this one: What is the simplified form of e^x / e^-3x ? I just confuse...

1. anonymous

the answer is e^(4x) right? :)

2. Owlcoffee

There is a law of exponents that states something useful: $\frac{ a^b }{ a^c }=a ^{b-c}$ So, translating it to the problem in question: $\frac{ e^x }{ e ^{-3x} }=e ^{x-(-3x)}=e ^{x +3x}$

3. Owlcoffee

We can easily prove it by stating the generic division of two exponential numbers: $\frac{ A ^{y} }{ A ^{x} }$ Let's suppose that $$y>x$$ and both belong to the real numbers, so therefore, we can, by definition: $\frac{ A.A.A.A.A...(y-factors) }{ A.A.A.A.A... (x-factors) }$ Since we have more y factors that x- factors, we will write them like: $(\frac{ A.A.A.A...(x-factors) }{ A.A.A.A...(x-factors) })((y-x)factors)$ Since the x factors divided x factors are just 1, we remain with the (y-x) factors. $A.A.A.A...(\left[ y-x \right] factors)$ And by definition of exponents: $A ^{y-x}$ So, in conclusion, we have just proven that: $\frac{ A^y}{ A^x }=A ^{y-x}$ Which is a property of the exponents.

4. anonymous

thanks.. :)