If `(16 ^(3x))(32 ^(x))(8 ^(3x))=((4 ^(6x)) (32^(3x)))/(4),` then what is the value of x ?

Options

A. `-2`
B. `-1`
C. 1
D. 2

Answer

To solve the equation \((16^{3x})(32^{x})(8^{3x}) = \frac{(4^{6x})(32^{3x})}{4}\), we will first express all numbers in terms of base 2.

### Step 1: Rewrite the bases in terms of 2
- \(16 = 2^4\)
- \(32 = 2^5\)
- \(8 = 2^3\)
- \(4 = 2^2\)

Now we can rewrite the left-hand side:
\[ (16^{3x}) = (2^4)^{3x} = 2^{12x} \]
\[ (32^{x}) = (2^5)^{x} = 2^{5x} \]
\[ (8^{3x}) = (2^3)^{3x} = 2^{9x} \]

Thus, the left-hand side becomes:
\[ 2^{12x} \cdot 2^{5x} \cdot 2^{9x} \]

### Step 2: Combine the exponents on the left-hand side
Using the property of exponents \(a^m \cdot a^n = a^{m+n}\):
\[ 2^{12x + 5x + 9x} = 2^{26x} \]

### Step 3: Rewrite the right-hand side
Now for the right-hand side:
\[ \frac{(4^{6x})(32^{3x})}{4} = \frac{(2^2)^{6x} \cdot (2^5)^{3x}}{2^2} \]
Calculating each part:
\[ (4^{6x}) = (2^2)^{6x} = 2^{12x} \]
\[ (32^{3x}) = (2^5)^{3x} = 2^{15x} \]

So the right-hand side becomes:
\[ \frac{2^{12x} \cdot 2^{15x}}{2^2} \]

### Step 4: Combine the exponents on the right-hand side
First, combine the exponents in the numerator:
\[ 2^{12x + 15x} = 2^{27x} \]
Now divide by \(2^2\):
\[ \frac{2^{27x}}{2^2} = 2^{27x - 2} \]

### Step 5: Set the exponents equal
Now we have:
\[ 2^{26x} = 2^{27x - 2} \]

Since the bases are the same, we can set the exponents equal to each other:
\[ 26x = 27x - 2 \]

### Step 6: Solve for x
Rearranging the equation gives:
\[ 27x - 26x = 2 \]
\[ x = 2 \]

### Final Answer
The value of \(x\) is \(2\).

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