If `root3(x/729)+root3((27x)/3375)=1`, then find the value of x.

Options

A. `(79507)/(3375)`
B. `(91125)/(2744)`
C. `(2025)/(196)`
D. `(443)/(125)`

Answer

To solve the equation \( \sqrt[3]{\frac{x}{729}} + \sqrt[3]{\frac{27x}{3375}} = 1 \), we will follow these steps:

### Step 1: Rewrite the equation
We start with the equation:
\[ \sqrt[3]{\frac{x}{729}} + \sqrt[3]{\frac{27x}{3375}} = 1 \]

### Step 2: Factor out the cube roots
We can factor out \( \sqrt[3]{x} \) from both terms:
\[ \sqrt[3]{x} \left( \sqrt[3]{\frac{1}{729}} + \sqrt[3]{\frac{27}{3375}} \right) = 1 \]

### Step 3: Simplify the cube roots
Next, we simplify the cube roots:
- \( \sqrt[3]{729} = 9 \) because \( 9^3 = 729 \)
- \( \sqrt[3]{3375} = 15 \) because \( 15^3 = 3375 \)
- \( \sqrt[3]{27} = 3 \) because \( 3^3 = 27 \)

Thus, we have:
\[ \sqrt[3]{\frac{1}{729}} = \frac{1}{9}, \quad \sqrt[3]{\frac{27}{3375}} = \frac{3}{15} = \frac{1}{5} \]

### Step 4: Substitute back into the equation
Now substituting these values back into the equation gives us:
\[ \sqrt[3]{x} \left( \frac{1}{9} + \frac{1}{5} \right) = 1 \]

### Step 5: Find a common denominator
To add \( \frac{1}{9} \) and \( \frac{1}{5} \), we find a common denominator, which is 45:
\[ \frac{1}{9} = \frac{5}{45}, \quad \frac{1}{5} = \frac{9}{45} \]
Thus,
\[ \frac{1}{9} + \frac{1}{5} = \frac{5}{45} + \frac{9}{45} = \frac{14}{45} \]

### Step 6: Substitute back into the equation
Now we substitute back:
\[ \sqrt[3]{x} \cdot \frac{14}{45} = 1 \]

### Step 7: Solve for \( \sqrt[3]{x} \)
To isolate \( \sqrt[3]{x} \), we multiply both sides by \( \frac{45}{14} \):
\[ \sqrt[3]{x} = \frac{45}{14} \]

### Step 8: Cube both sides to find \( x \)
Now we cube both sides to find \( x \):
\[ x = \left( \frac{45}{14} \right)^3 = \frac{45^3}{14^3} \]
Calculating \( 45^3 \) and \( 14^3 \):
\[ 45^3 = 91125, \quad 14^3 = 2744 \]
Thus,
\[ x = \frac{91125}{2744} \]

### Final Answer
The value of \( x \) is:
\[ x = \frac{91125}{2744} \]