Find the cube-roots of :
`729xx8000`

Answer

To find the cube roots of \( 729 \times 8000 \), we can follow these steps:

### Step 1: Factor the numbers
First, we need to factor both \( 729 \) and \( 8000 \).

- **Factoring 729:**
\[ 729 = 3 \times 243 \]
\[ 243 = 3 \times 81 \]
\[ 81 = 3 \times 27 \]
\[ 27 = 3 \times 9 \]
\[ 9 = 3 \times 3 \]
Thus, \( 729 = 3^6 \).

- **Factoring 8000:**
\[ 8000 = 8 \times 1000 \]
\[ 8 = 2^3 \]
\[ 1000 = 10^3 = (2 \times 5)^3 = 2^3 \times 5^3 \]
Therefore, \( 8000 = 2^3 \times (2^3 \times 5^3) = 2^6 \times 5^3 \).

### Step 2: Combine the factors
Now, we can combine the factors of \( 729 \) and \( 8000 \):
\[ 729 \times 8000 = 3^6 \times (2^6 \times 5^3) = 3^6 \times 2^6 \times 5^3 \]

### Step 3: Find the cube root
To find the cube root of \( 729 \times 8000 \), we apply the cube root to each factor:
\[ \sqrt[3]{729 \times 8000} = \sqrt[3]{3^6} \times \sqrt[3]{2^6} \times \sqrt[3]{5^3} \]

Using the property of exponents:
\[ \sqrt[3]{3^6} = 3^{6/3} = 3^2 = 9 \]
\[ \sqrt[3]{2^6} = 2^{6/3} = 2^2 = 4 \]
\[ \sqrt[3]{5^3} = 5^{3/3} = 5^1 = 5 \]

### Step 4: Multiply the results
Now, we multiply the results:
\[ 9 \times 4 \times 5 = 36 \times 5 = 180 \]

### Final Answer
Thus, the cube root of \( 729 \times 8000 \) is \( 180 \).