SOLUTION: I need help with this problem. Thank you. Write the given function in the form y=ab^x {{{y= sqrt(9^(x-1))}}}

Algebra ->  Square-cubic-other-roots -> SOLUTION: I need help with this problem. Thank you. Write the given function in the form y=ab^x {{{y= sqrt(9^(x-1))}}}      Log On


   

Question 423081: I need help with this problem. Thank you.
Write the given function in the form y=ab^x
y=+sqrt%289%5E%28x-1%29%29
Answer by jsmallt9(3759) About Me  (Show Source):
You can put this solution on YOUR website!
y=+sqrt%289%5E%28x-1%29%29
Since the desired form has no square roots, we will start by eliminating the square root. We can replace the square root with an exponent of 1/2. Because of occasional display problems with fractional exponents I am going to use the decimal equivalent of 1/2, 0.5:
y=+%289%5E%28x-1%29%29%5E0.5
We now have a power of a power of 8. The rule for exponents for this is to multiply the exponents:
y=++9%5E%28%28x-1%29%2A0.5%29
We still want an exponent of just x. So we need still to eliminate the -1 and the 0.5. Since 9+=+3%5E2 we can introduce a 2 into the exponent:
y=++%283%5E2%29%5E%28%28x-1%29%2A0.5%29
Again we use the rule for exponents:
y=++3%5E%282%2A%28x-1%29%2A0.5%29
The exponent simplifies to:
y=++3%5E%28x-1%29
We're making progress. All we have to do now is eliminate the -1. The exponent is now a subtraction. And when do we subtract exponents? Answer: When we divide. So if we "undo" a division involving 3%5Ex and 3%5E1...
y=++3%5Ex%2F3%5E1
The denominator simplifies:
y=++3%5Ex%2F3
The exponent is finally what we want it to be. But we want something times 3%5Ex not 3%5Ex divided by something. Since division by 3 is the same as multiplying by the reciprocal of 3 this is an easy checge:
y=++%281%2F3%293%5Ex
We finally have the desired form with the "a" being 1/3 and the "b" being 3.

P.S. In response to the question in your "Thank you"...
The problem asked for an equation of the form:
y+=+a%2Ab%5Ex
With the steps I've shown above we've transformed
y=+sqrt%289%5E%28x-1%29%29
into
y+=+%281%2F3%29%2A3%5Ex
which is the desired form. So I don't understand your question: "Does x stay the exponent?" First of all, the desired form wants the exponent to be x so we don't want the exponent to stop being x. Second, I'm not sure where the "x" can go. (Remember, the order of operations (aka PEMDAS) requires that we raise 3 to the x power before we multiply by 1/3. So the 1/3 and the 3 do not cancel.)