Find the cube-roots of :
`-15.625`

Answer

To find the cube root of \(-15.625\), we can follow these steps:

### Step 1: Rewrite the number
First, we can express \(-15.625\) as a fraction. This can be written as:
\[ -15.625 = -\frac{15625}{1000} \]

### Step 2: Factor the numerator
Next, we need to factor \(15625\). We can start dividing by \(5\) (since \(15625\) ends in \(5\)):
\[ 15625 \div 5 = 3125 \]
Continuing to divide by \(5\):
\[ 3125 \div 5 = 625 \]
\[ 625 \div 5 = 125 \]
\[ 125 \div 5 = 25 \]
\[ 25 \div 5 = 5 \]
\[ 5 \div 5 = 1 \]
So, we have:
\[ 15625 = 5^6 \]

### Step 3: Rewrite the fraction
Now, substituting back, we have:
\[ -15.625 = -\frac{5^6}{10^3} \]

### Step 4: Simplify the cube root
We can now take the cube root of both the numerator and the denominator:
\[ \sqrt[3]{-15.625} = \sqrt[3]{-\frac{5^6}{10^3}} = \frac{\sqrt[3]{-5^6}}{\sqrt[3]{10^3}} \]

### Step 5: Calculate the cube roots
The cube root of \(-5^6\) is:
\[ \sqrt[3]{-5^6} = -5^2 = -25 \]
And the cube root of \(10^3\) is:
\[ \sqrt[3]{10^3} = 10 \]

### Step 6: Final calculation
Putting it all together:
\[ \sqrt[3]{-15.625} = \frac{-25}{10} = -2.5 \]

Thus, the cube root of \(-15.625\) is:
\[ \boxed{-2.5} \]

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