Evaluate : `root3(1728/2744)`

Options

A. `6/11`
B. `6/7`
C. `3/4`
D. `12/17`

Answer

To evaluate \( \sqrt[3]{\frac{1728}{2744}} \), we can follow these steps:

### Step 1: Write the expression
We start with the expression:
\[ \sqrt[3]{\frac{1728}{2744}} \]

### Step 2: Simplify the fraction
Next, we simplify the fraction \( \frac{1728}{2744} \). We can divide both the numerator and the denominator by their greatest common divisor (GCD).

First, we can find the prime factorization of both numbers:
- \( 1728 = 12^3 = (2^2 \cdot 3)^3 = 2^6 \cdot 3^3 \)
- \( 2744 = 14^3 = (2 \cdot 7)^3 = 2^3 \cdot 7^3 \)

Now, we can write:
\[ \frac{1728}{2744} = \frac{2^6 \cdot 3^3}{2^3 \cdot 7^3} \]

### Step 3: Cancel common factors
Now we can simplify the fraction:
\[ \frac{2^6 \cdot 3^3}{2^3 \cdot 7^3} = \frac{2^{6-3} \cdot 3^3}{7^3} = \frac{2^3 \cdot 3^3}{7^3} = \frac{216}{343} \]

### Step 4: Rewrite the cube root
Now we can rewrite the expression:
\[ \sqrt[3]{\frac{1728}{2744}} = \sqrt[3]{\frac{216}{343}} \]

### Step 5: Evaluate the cube roots
Next, we can evaluate the cube roots of the numerator and the denominator:
- The cube root of \( 216 \) is \( 6 \) (since \( 6^3 = 216 \)).
- The cube root of \( 343 \) is \( 7 \) (since \( 7^3 = 343 \)).

Thus, we have:
\[ \sqrt[3]{\frac{216}{343}} = \frac{\sqrt[3]{216}}{\sqrt[3]{343}} = \frac{6}{7} \]

### Final Answer
Therefore, the final answer is:
\[ \sqrt[3]{\frac{1728}{2744}} = \frac{6}{7} \]

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