`sqrt(2^x(4^x(0. 125)^(1/ x))^(1/3))=4.(2^(1/3))`

Options

A. `(14)/(5)`
B. 3
C. `-(1)/(5)`
D. `-(3)/(5)`

Answer

To solve the equation \( \sqrt{2^x (4^x (0.125)^{(1/x)})^{(1/3)}} = 4 \cdot (2^{1/3}) \), we will follow these steps:

### Step 1: Rewrite the equation
First, we rewrite \( 4^x \) and \( 0.125 \) in terms of base 2:
- \( 4^x = (2^2)^x = 2^{2x} \)
- \( 0.125 = \frac{1}{8} = \frac{1}{2^3} = 2^{-3} \)

Thus, we can rewrite the equation as:
\[ \sqrt{2^x \left(2^{2x} \left(2^{-3}\right)^{(1/x)}\right)^{(1/3)}} = 4 \cdot (2^{1/3}) \]

### Step 2: Simplify the left side
Now, simplify the left side:
\[ \sqrt{2^x \left(2^{2x} \cdot 2^{-3/x}\right)^{(1/3)}} \]

Combine the exponents inside the square root:
\[ = \sqrt{2^x \cdot 2^{(2x - 3/x)/3}} = \sqrt{2^{x + (2x - 3/x)/3}} \]

### Step 3: Combine exponents
The exponent can be combined:
\[ = \sqrt{2^{\frac{3x + 2x - 3/x}{3}}} = \sqrt{2^{\frac{5x - 3/x}{3}}} \]

### Step 4: Apply the square root
Using the property of square roots:
\[ = 2^{\frac{5x - 3/x}{6}} \]

### Step 5: Simplify the right side
Now simplify the right side:
\[ 4 \cdot (2^{1/3}) = 2^2 \cdot 2^{1/3} = 2^{2 + 1/3} = 2^{\frac{6}{3} + \frac{1}{3}} = 2^{\frac{7}{3}} \]

### Step 6: Set the exponents equal
Now we can set the exponents equal to each other:
\[ \frac{5x - 3/x}{6} = \frac{7}{3} \]

### Step 7: Clear the fraction
Multiply both sides by 6:
\[ 5x - \frac{3}{x} = 14 \]

### Step 8: Rearrange the equation
Rearranging gives:
\[ 5x - 14 - \frac{3}{x} = 0 \]

### Step 9: Multiply through by \( x \)
To eliminate the fraction, multiply through by \( x \):
\[ 5x^2 - 14x - 3 = 0 \]

### Step 10: Solve the quadratic equation
Now we can use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
- Here, \( a = 5 \), \( b = -14 \), and \( c = -3 \).

Calculating the discriminant:
\[ b^2 - 4ac = (-14)^2 - 4 \cdot 5 \cdot (-3) = 196 + 60 = 256 \]

Now substituting into the formula:
\[ x = \frac{14 \pm \sqrt{256}}{10} = \frac{14 \pm 16}{10} \]

### Step 11: Find the values of \( x \)
Calculating the two possible values:
1. \( x = \frac{30}{10} = 3 \)
2. \( x = \frac{-2}{10} = -0.2 \)

### Final Solution
Thus, the solutions are:
\[ x = 3 \quad \text{and} \quad x = -0.2 \]