Find the cube-roots of :
`125/216`

Answer

To find the cube root of \( \frac{125}{216} \), we can use the prime factorization method. Here’s a step-by-step solution:

### Step 1: Prime Factorization of the Numerator and Denominator
- **Numerator (125)**:
- \( 125 = 5 \times 25 \)
- \( 25 = 5 \times 5 \)
- Therefore, \( 125 = 5^3 \).

- **Denominator (216)**:
- \( 216 = 2 \times 108 \)
- \( 108 = 2 \times 54 \)
- \( 54 = 2 \times 27 \)
- \( 27 = 3 \times 9 \)
- \( 9 = 3 \times 3 \)
- Therefore, \( 216 = 2^3 \times 3^3 \).

### Step 2: Write the Fraction with Prime Factors
Now we can write the fraction using its prime factors:
\[ \frac{125}{216} = \frac{5^3}{2^3 \times 3^3} \]

### Step 3: Apply the Cube Root
To find the cube root of the fraction, we can apply the cube root to both the numerator and the denominator:
\[ \sqrt[3]{\frac{125}{216}} = \frac{\sqrt[3]{125}}{\sqrt[3]{216}} = \frac{\sqrt[3]{5^3}}{\sqrt[3]{2^3 \times 3^3}} \]

### Step 4: Simplify the Cube Roots
- The cube root of \( 5^3 \) is \( 5 \).
- The cube root of \( 2^3 \) is \( 2 \).
- The cube root of \( 3^3 \) is \( 3 \).

Thus, we have:
\[ \sqrt[3]{\frac{125}{216}} = \frac{5}{2 \times 3} = \frac{5}{6} \]

### Final Answer
The cube root of \( \frac{125}{216} \) is \( \frac{5}{6} \).

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