Find the cube-roots of :
`-27/125`

Answer

To find the cube root of \(-\frac{27}{125}\), we can follow these steps:

### Step 1: Identify the expression
We start with the expression:
\[ a = -\frac{27}{125} \]

### Step 2: Express the cube root
We want to find:
\[ a^{\frac{1}{3}} = \left(-\frac{27}{125}\right)^{\frac{1}{3}} \]

### Step 3: Factor the numerator and denominator
Next, we can factor the numerator and denominator:
\[ -\frac{27}{125} = -\frac{3^3}{5^3} \]

### Step 4: Rewrite the expression
Now we rewrite the expression using the factors:
\[ \left(-\frac{3^3}{5^3}\right)^{\frac{1}{3}} \]

### Step 5: Apply the property of exponents
Using the property of exponents \((a^m)^{n} = a^{m \cdot n}\), we can simplify:
\[ \left(-3^3\right)^{\frac{1}{3}} \div \left(5^3\right)^{\frac{1}{3}} = \frac{-3^{3 \cdot \frac{1}{3}}}{5^{3 \cdot \frac{1}{3}}} \]

### Step 6: Simplify the expression
This simplifies to:
\[ \frac{-3^1}{5^1} = \frac{-3}{5} \]

### Final Answer
Thus, the cube root of \(-\frac{27}{125}\) is:
\[ -\frac{3}{5} \]

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