Find x, if :
`4^(2x) = (1)/(32)`

Answer

To solve the equation \( 4^{2x} = \frac{1}{32} \), we will follow these steps:

### Step 1: Rewrite the bases
First, we need to express both sides of the equation with the same base. We know that \( 4 \) can be rewritten as \( 2^2 \) and \( 32 \) can be rewritten as \( 2^5 \).

So, we can rewrite the left side:
\[ 4^{2x} = (2^2)^{2x} \]
Using the power of a power property, we can simplify this to:
\[ (2^2)^{2x} = 2^{4x} \]

Now, we can rewrite the right side:
\[ \frac{1}{32} = \frac{1}{2^5} = 2^{-5} \]

### Step 2: Set the exponents equal
Now we have:
\[ 2^{4x} = 2^{-5} \]
Since the bases are the same, we can set the exponents equal to each other:
\[ 4x = -5 \]

### Step 3: Solve for \( x \)
Now, we can solve for \( x \) by dividing both sides by \( 4 \):
\[ x = \frac{-5}{4} \]

### Final Answer
Thus, the solution is:
\[ x = -\frac{5}{4} \]

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