If `4^(x+3)=112+8xx4^(x)`, find `(18x)^(3x)`

Answer

To solve the equation \( 4^{(x+3)} = 112 + 8 \cdot 4^x \), and find \( (18x)^{3x} \), we can follow these steps:

### Step 1: Rewrite the equation
Start with the given equation:
\[ 4^{(x+3)} = 112 + 8 \cdot 4^x \]

### Step 2: Simplify the left side
Using the property of exponents, we can rewrite \( 4^{(x+3)} \) as:
\[ 4^{(x+3)} = 4^x \cdot 4^3 = 4^x \cdot 64 \]
So, the equation becomes:
\[ 64 \cdot 4^x = 112 + 8 \cdot 4^x \]

### Step 3: Rearrange the equation
Now, we can move the \( 8 \cdot 4^x \) term to the left side:
\[ 64 \cdot 4^x - 8 \cdot 4^x = 112 \]
This simplifies to:
\[ (64 - 8) \cdot 4^x = 112 \]
\[ 56 \cdot 4^x = 112 \]

### Step 4: Solve for \( 4^x \)
Now, divide both sides by 56:
\[ 4^x = \frac{112}{56} \]
\[ 4^x = 2 \]

### Step 5: Express \( 4^x \) in terms of base 2
Since \( 4 = 2^2 \), we can write:
\[ (2^2)^x = 2 \]
This simplifies to:
\[ 2^{2x} = 2^1 \]

### Step 6: Set the exponents equal
Since the bases are the same, we can set the exponents equal to each other:
\[ 2x = 1 \]
Now, solve for \( x \):
\[ x = \frac{1}{2} \]

### Step 7: Find \( (18x)^{3x} \)
Now that we have \( x \), we can substitute it into \( (18x)^{3x} \):
\[ (18 \cdot \frac{1}{2})^{3 \cdot \frac{1}{2}} = (9)^{\frac{3}{2}} \]

### Step 8: Simplify \( 9^{\frac{3}{2}} \)
We can express \( 9 \) as \( 3^2 \):
\[ (3^2)^{\frac{3}{2}} = 3^{2 \cdot \frac{3}{2}} = 3^3 \]
Calculating \( 3^3 \):
\[ 3^3 = 27 \]

### Final Answer
Thus, the value of \( (18x)^{3x} \) is:
\[ \boxed{27} \]