Find the value of `3sqrt((0.000729)/(0.085184))`.

Options

A. `(9)/(44)`
B. `(7)/(44)`
C. `(3)/(44)`
D. `(6)/(44)`

Answer

To solve the problem \(3\sqrt{\frac{0.000729}{0.085184}}\), we will follow these steps:

### Step 1: Remove the Decimals
First, we need to eliminate the decimals in the fraction. We can do this by multiplying both the numerator and the denominator by \(10^6\) (which is 1,000,000) since both numbers have 6 decimal places.

\[ 3\sqrt{\frac{0.000729 \times 10^6}{0.085184 \times 10^6}} = 3\sqrt{\frac{729}{85184}} \]

### Step 2: Simplify the Numerator
Next, we simplify the numerator. We know that \(729\) is a perfect cube:

\[ 729 = 9^3 \]

### Step 3: Factor the Denominator
Now, we need to factor \(85184\) to see if it can be expressed as a cube. We will start by dividing by small prime numbers.

1. Divide by \(4\):
\[ 85184 \div 4 = 21296 \]
2. Divide by \(4\) again:
\[ 21296 \div 4 = 5324 \]
3. Divide by \(4\) again:
\[ 5324 \div 4 = 1331 \]
4. Now, \(1331\) can be factored further:
\[ 1331 = 11^3 \]

So, we can express \(85184\) as:
\[ 85184 = 4^3 \times 11^3 \]

### Step 4: Rewrite the Expression
Now we can rewrite the expression under the cube root:

\[ 3\sqrt{\frac{9^3}{4^3 \times 11^3}} = 3\sqrt{\frac{9^3}{(4 \times 11)^3}} = 3\sqrt{\frac{9^3}{44^3}} \]

### Step 5: Simplify the Cube Root
Using the property of cube roots, we can simplify:

\[ 3\sqrt{\frac{9^3}{44^3}} = 3 \cdot \frac{9}{44} \]

### Step 6: Final Calculation
Now we can multiply:

\[ 3 \cdot \frac{9}{44} = \frac{27}{44} \]

### Final Answer
Thus, the final value of \(3\sqrt{\frac{0.000729}{0.085184}}\) is:

\[ \frac{27}{44} \]

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