If r and s are positive integers and 8^r=4^s, what is the value of s/r

Question

anonymous · Answer

THIS IS SO HARD

anonymous · Answer

Have you studied logarithms?

anonymous · Answer

no

anonymous · Answer

Okay. Let's take a simpler approach. 

We know that $$8 = 2^3 ~~\rm{and} ~~ 4 = 2^2$$

Therefore, we can rewrite the expression as$$2^{3r} = 2^{2s}$$

anonymous · Answer

okay

anonymous · Answer

Do you follow how I went from $8^r$ to $2^{3r}$?

anonymous · Answer

No c

anonymous · Answer

Okay, Recall that$$a^c \cdot a^d = a^{c \cdot d}$$

anonymous · Answer

oh yeah im starting to remember

anonymous · Answer

Sorry. That's wrong. 

$$a^c \cdot a^d = {a^{c + d}}$$

We need $$(a^c)^d = a^{c \cdot d}$$

anonymous · Answer

its alright

anonymous · Answer

If we have $8^r$ and we substitute $2^3$ in for 8, we get $$\left ( 2^3 \right ) ^r$$This simplifies to $$2^{3r}$$

anonymous · Answer

Now, by Algebra magic, we can write $$2^{3r} = 2^{2s}$$ as$$3r = 2s$$

Can you find s/r now?

anonymous · Answer

what happens to the two big numbers

anonymous · Answer

I forget the proof, but let me try to explain. 

Let's say we have $$y^x = y^z$$Since y has the same value on both sides of the equal sign, we know that for $$y^x = y^z$$ to be true, $$x = z$$ has to be true. 

The same applies here, y = 2, x = 3r, and z = 2s

3r = 2s

anonymous · Answer

after 3r=2s what happens next

anonymous · Answer

Set up the ratio of s/r$${s \over r} = {3 \over 2}$$If we cross-multiply, we end up back at 3r = 2s

anonymous · Answer

i see.and when we do cross-multiply what happens next. is 3r=2s the answer

anonymous · Answer

No. We want the ratio s/r. I was just noting how to check to make sure we had the right ratio. If we cross multiply and get the original expression, we are right. 
The answer is s/r = 3/2

anonymous · Answer

ohh okay thank you so much