Find the cube-roots of :
`64xx27`

Answer

To find the cube root of the product \( 64 \times 27 \), we can follow these steps:

### Step 1: Define the expression
Let \( A = 64 \times 27 \).

### Step 2: Write the cube root expression
We want to find \( A^{1/3} \), which is \( (64 \times 27)^{1/3} \).

### Step 3: Break down the numbers into their prime factors
We can express \( 64 \) and \( 27 \) in terms of their prime factors:
- \( 64 = 4^3 \) (since \( 4 \times 4 \times 4 = 64 \))
- \( 27 = 3^3 \) (since \( 3 \times 3 \times 3 = 27 \))

### Step 4: Substitute the prime factors into the expression
Now, we can rewrite the expression:
\[ (64 \times 27)^{1/3} = (4^3 \times 3^3)^{1/3} \]

### Step 5: Apply the property of exponents
Using the property \( (a^m \times b^m) = (a \times b)^m \), we can combine the bases:
\[ (4^3 \times 3^3)^{1/3} = (4 \times 3)^{3 \times \frac{1}{3}} = (4 \times 3)^1 \]

### Step 6: Calculate the product
Now we calculate \( 4 \times 3 \):
\[ 4 \times 3 = 12 \]

### Step 7: Final answer
Thus, the cube root of \( 64 \times 27 \) is:
\[ \sqrt[3]{64 \times 27} = 12 \]

### Summary
The cube root of \( 64 \times 27 \) is \( 12 \).

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