Find the cube-roots of :
`9.261`

Answer

To find the cube root of \( 9.261 \), we will follow these steps:

### Step 1: Rewrite the number
We can express \( 9.261 \) as a fraction:
\[ 9.261 = \frac{9261}{1000} \]

### Step 2: Prime factorization of the numerator
Next, we need to find the prime factorization of \( 9261 \).

- Check divisibility by \( 2 \): \( 9261 \) is not even, so it is not divisible by \( 2 \).
- Check divisibility by \( 3 \): The sum of the digits \( 9 + 2 + 6 + 1 = 18 \) is divisible by \( 3 \). Thus, \( 9261 \) is divisible by \( 3 \).

Perform the division:
\[ 9261 \div 3 = 3087 \]
Continuing with \( 3087 \):
\[ 3087 \div 3 = 1029 \]
Continuing with \( 1029 \):
\[ 1029 \div 3 = 343 \]
Now, \( 343 \) is not divisible by \( 3 \), but it is \( 7^3 \):
\[ 343 = 7 \times 7 \times 7 \]

So, the prime factorization of \( 9261 \) is:
\[ 9261 = 3^3 \times 7^3 \]

### Step 3: Prime factorization of the denominator
Now, we factor \( 1000 \):
\[ 1000 = 10^3 = (2 \times 5)^3 = 2^3 \times 5^3 \]

### Step 4: Combine the factors
Now we can combine the factors:
\[ 9.261 = \frac{3^3 \times 7^3}{2^3 \times 5^3} \]

### Step 5: Finding the cube root
To find the cube root, we take the cube root of the numerator and the denominator separately:
\[ \sqrt[3]{9.261} = \frac{\sqrt[3]{3^3 \times 7^3}}{\sqrt[3]{2^3 \times 5^3}} = \frac{3 \times 7}{2 \times 5} \]

Calculating this gives:
\[ \frac{21}{10} = 2.1 \]

### Final Answer
Thus, the cube root of \( 9.261 \) is:
\[ \boxed{2.1} \]