If `log_(6)log_(2)[sqrt(4x+2)+2sqrtx]=0thenx=____.`

Answer

To solve the equation \( \log_{6}(\log_{2}(\sqrt{4x+2} + 2\sqrt{x})) = 0 \), we will follow these steps:

### Step 1: Apply the property of logarithms
Since \( \log_{6}(y) = 0 \) implies \( y = 6^0 = 1 \), we can rewrite the equation as:
\[ \log_{2}(\sqrt{4x + 2} + 2\sqrt{x}) = 1 \]

**Hint:** Remember that \( \log_{a}(b) = c \) implies \( b = a^c \).

### Step 2: Solve for the inner logarithm
Now, we can convert the logarithmic equation to its exponential form:
\[ \sqrt{4x + 2} + 2\sqrt{x} = 2^1 = 2 \]

**Hint:** The base of the logarithm tells us what to raise to the power of the result.

### Step 3: Rearrange the equation
Next, isolate the square root term:
\[ \sqrt{4x + 2} = 2 - 2\sqrt{x} \]

**Hint:** To isolate a term, you can move other terms to the opposite side of the equation.

### Step 4: Square both sides
Now, square both sides to eliminate the square root:
\[ 4x + 2 = (2 - 2\sqrt{x})^2 \]

**Hint:** When squaring a binomial, remember to apply the formula \( (a - b)^2 = a^2 - 2ab + b^2 \).

### Step 5: Expand the right side
Expanding the right side gives:
\[ 4x + 2 = 4 - 8\sqrt{x} + 4x \]

**Hint:** Carefully distribute each term when expanding.

### Step 6: Simplify the equation
Now, simplify the equation by subtracting \( 4x \) from both sides:
\[ 2 = 4 - 8\sqrt{x} \]

**Hint:** Combine like terms to simplify the equation.

### Step 7: Isolate the square root
Rearranging gives:
\[ 8\sqrt{x} = 4 - 2 \]
\[ 8\sqrt{x} = 2 \]

**Hint:** Always aim to isolate the variable or the term involving the variable.

### Step 8: Solve for \( \sqrt{x} \)
Dividing both sides by 8:
\[ \sqrt{x} = \frac{2}{8} = \frac{1}{4} \]

**Hint:** To isolate the square root, you can divide both sides by the coefficient.

### Step 9: Square both sides again
Now, square both sides to find \( x \):
\[ x = \left(\frac{1}{4}\right)^2 = \frac{1}{16} \]

**Hint:** Squaring a fraction involves squaring both the numerator and the denominator.

### Final Answer
Thus, the value of \( x \) is:
\[ \boxed{\frac{1}{16}} \]