Quote: Solve: (1/3)^x = sqrt(27^6)
I can't find past complete examples in my book & I'm studying for my exam... any help would be appreciated!
It's important to note what a square root represents.
A 'square root' is the same thing as the exponent (1/2)!
Let us rewrite the above equation:
(1/3)^x = (27^6)^(1/2)
(1/3)^x = 27^3
The left side is a fraction. To make it a whole number, the power must be negative ( x < 1 ).
Now we need to determine what that exponent is:
(1/27)^x = 27^9
(27/1)^-9 = 27^9
Therefore, x = -9
That way, with numbers as simple as this, you can really think it through (you don't really need to do all the steps, you can probably do it in your head, but with different numbers you might have to do some more work.
There's a much easier way to solve this problem, and that is using Logs.
If you don't know what a Logarithm is, it's an exponential scale (instead of counting like 1, 2, 3, 4 or like 10, 20, 30, 40, you are instead going like 10^1, 10^2, 10^3, 10^4)
Logs allow you to work directly with the exponents.
To solve this with logs we would do this:
(1/3)^x = (27^6)^(1/2)
(1/3)^x = 27^3
Log1/3(27^3) = x
x = -9
+30 
