Which equation is equivalent to (1/3)^x = 27 ^x + 2

The Correct Answer and Explanation is:

We are tasked with solving for ( x ) in the equation:

[
\left(\frac{1}{3}\right)^x = 27^{x+2}
]

Step 1: Express everything with a common base

To solve this equation, we should first express both sides using the same base. We observe that both ( \frac{1}{3} ) and ( 27 ) are powers of 3:

( \frac{1}{3} = 3^{-1} )
( 27 = 3^3 )

Now we can rewrite the equation as:

[
(3^{-1})^x = (3^3)^{x+2}
]

Step 2: Simplify the exponents

Next, simplify both sides by applying the rule of exponents ( (a^m)^n = a^{m \cdot n} ).

On the left-hand side: ( (3^{-1})^x = 3^{-x} )
On the right-hand side: ( (3^3)^{x+2} = 3^{3(x+2)} = 3^{3x+6} )

So the equation becomes:

[
3^{-x} = 3^{3x + 6}
]

Step 3: Set the exponents equal

Since the bases are the same (both are powers of 3), we can set the exponents equal to each other:

[
-x = 3x + 6
]

Step 4: Solve for ( x )

Now solve for ( x ):

Add ( x ) to both sides:

[
0 = 4x + 6
]

Subtract 6 from both sides:

[
-6 = 4x
]

Divide by 4:

[
x = -\frac{3}{2}
]

Final Answer:

The solution to the equation is: