Find the value of `(27 xx 2744)^(1//3)`

Answer

To solve the expression \((27 \times 2744)^{\frac{1}{3}}\), we will break it down step by step.

### Step 1: Factorize 27
We know that:
\[ 27 = 3^3 \]

### Step 2: Factorize 2744
Next, we need to factorize 2744. We can do this by dividing by prime numbers:
- 2744 is even, so we divide by 2:
\[ 2744 \div 2 = 1372 \]
- 1372 is even, divide by 2 again:
\[ 1372 \div 2 = 686 \]
- 686 is even, divide by 2 again:
\[ 686 \div 2 = 343 \]
- 343 is \(7^3\) (since \(7 \times 7 \times 7 = 343\)).

Thus, we can write:
\[ 2744 = 2^3 \times 7^3 \]

### Step 3: Combine the factorizations
Now we can combine the factorizations of both numbers:
\[ 27 \times 2744 = 3^3 \times (2^3 \times 7^3) = 3^3 \times 2^3 \times 7^3 \]

### Step 4: Rewrite the expression
Now we substitute this back into our expression:
\[ (27 \times 2744)^{\frac{1}{3}} = (3^3 \times 2^3 \times 7^3)^{\frac{1}{3}} \]

### Step 5: Apply the power of a product rule
Using the property \((a \times b)^c = a^c \times b^c\), we can simplify:
\[ (3^3)^{\frac{1}{3}} \times (2^3)^{\frac{1}{3}} \times (7^3)^{\frac{1}{3}} = 3^{3 \times \frac{1}{3}} \times 2^{3 \times \frac{1}{3}} \times 7^{3 \times \frac{1}{3}} \]
This simplifies to:
\[ 3^1 \times 2^1 \times 7^1 = 3 \times 2 \times 7 \]

### Step 6: Calculate the final result
Now we calculate:
\[ 3 \times 2 = 6 \]
\[ 6 \times 7 = 42 \]

Thus, the value of the expression \((27 \times 2744)^{\frac{1}{3}}\) is:
\[ \boxed{42} \]