What is the value of `'x'" in "2 sqrt(3x)-5 sqrt(27 x)-sqrt(108x) = -19` ?

Options

A. `0.33`
B. 1
C. 3
D. 9

Answer

To solve the equation \( 2\sqrt{3x} - 5\sqrt{27x} - \sqrt{108x} = -19 \), we will follow these steps:

### Step 1: Simplify the square root terms
We start by simplifying the square root terms in the equation.

1. The term \( \sqrt{27x} \) can be rewritten as:
\[ \sqrt{27x} = \sqrt{9 \cdot 3x} = \sqrt{9} \cdot \sqrt{3x} = 3\sqrt{3x} \]
Thus, \( -5\sqrt{27x} = -5 \cdot 3\sqrt{3x} = -15\sqrt{3x} \).

2. The term \( \sqrt{108x} \) can be rewritten as:
\[ \sqrt{108x} = \sqrt{36 \cdot 3x} = \sqrt{36} \cdot \sqrt{3x} = 6\sqrt{3x} \]

Now substituting these back into the equation gives:
\[ 2\sqrt{3x} - 15\sqrt{3x} - 6\sqrt{3x} = -19 \]

### Step 2: Combine like terms
Now we combine the like terms on the left side:
\[ (2 - 15 - 6)\sqrt{3x} = -19 \]
Calculating the coefficients:
\[ -19\sqrt{3x} = -19 \]

### Step 3: Divide both sides by -19
To isolate \( \sqrt{3x} \), we divide both sides by -19:
\[ \sqrt{3x} = 1 \]

### Step 4: Square both sides
Next, we square both sides to eliminate the square root:
\[ 3x = 1^2 \]
Thus,
\[ 3x = 1 \]

### Step 5: Solve for \( x \)
Finally, we solve for \( x \) by dividing both sides by 3:
\[ x = \frac{1}{3} \]

### Final Answer
The value of \( x \) is \( \frac{1}{3} \) or approximately \( 0.333 \).

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