Evaluate : `root3(57(132)/343)`

Options

A. `27/7`
B. `23/7`
C. `23/5`
D. `27/8`

Answer

To evaluate the expression \( \sqrt[3]{\frac{57 \times 132}{343}} \), we can follow these steps:

### Step 1: Simplify the expression
First, we rewrite the expression:
\[ \sqrt[3]{\frac{57 \times 132}{343}} = \sqrt[3]{\frac{57 \times 132}{7^3}} \]

### Step 2: Calculate the numerator
Now, we need to calculate the numerator:
\[ 57 \times 132 \]
Calculating this gives:
\[ 57 \times 132 = 7524 \]

### Step 3: Rewrite the expression
Now we can rewrite the expression with the calculated numerator:
\[ \sqrt[3]{\frac{7524}{343}} = \sqrt[3]{\frac{7524}{7^3}} \]

### Step 4: Factor the numerator
Next, we need to factor \( 7524 \) to see if it can be expressed in terms of cubes. We can start by dividing it by 3:
\[ 7524 \div 3 = 2508 \]
Continuing to factor \( 2508 \):
\[ 2508 \div 3 = 836 \]
Next, factor \( 836 \):
\[ 836 \div 4 = 209 \]
And finally, \( 209 \) can be factored as:
\[ 209 = 11 \times 19 \]
Thus, we have:
\[ 7524 = 3^2 \times 4 \times 11 \times 19 \]

### Step 5: Rewrite the numerator in terms of cubes
We can express \( 4 \) as \( 2^2 \):
\[ 7524 = 3^2 \times (2^2) \times 11 \times 19 \]
We can see that \( 7524 \) does not have a complete cube factorization.

### Step 6: Calculate the cube root
Now we can rewrite the expression:
\[ \sqrt[3]{\frac{7524}{343}} = \sqrt[3]{\frac{3^2 \times 2^2 \times 11 \times 19}{7^3}} \]
This simplifies to:
\[ \frac{\sqrt[3]{3^2 \times 2^2 \times 11 \times 19}}{7} \]

### Step 7: Final expression
Thus, the final expression is:
\[ \frac{\sqrt[3]{3^2 \times 2^2 \times 11 \times 19}}{7} \]

### Final Answer
The evaluated expression is:
\[ \frac{\sqrt[3]{7524}}{7} \]