Find the cube-roots of :
8000

Answer

To find the cube root of 8000, we can follow these steps:

### Step 1: Write the expression for the cube root
We start with the number 8000 and express it as \( A \):
\[ A = 8000 \]
We want to find \( A^{1/3} \).

### Step 2: Rewrite 8000 in terms of its factors
We can factor 8000 into smaller components:
\[ 8000 = 8 \times 1000 \]

### Step 3: Factor each component further
Next, we can factor both 8 and 1000:
\[ 8 = 2^3 \quad \text{and} \quad 1000 = 10^3 \]
Thus, we can rewrite 8000 as:
\[ 8000 = 2^3 \times 10^3 \]

### Step 4: Apply the cube root
Now we can take the cube root of the entire expression:
\[ A^{1/3} = (2^3 \times 10^3)^{1/3} \]

### Step 5: Use the property of exponents
Using the property of exponents \( (a^m \times b^n)^{p} = a^{mp} \times b^{np} \), we can simplify:
\[ (2^3)^{1/3} \times (10^3)^{1/3} = 2^{3 \times \frac{1}{3}} \times 10^{3 \times \frac{1}{3}} = 2^1 \times 10^1 \]

### Step 6: Calculate the result
Now we can calculate:
\[ 2^1 \times 10^1 = 2 \times 10 = 20 \]

### Final Answer
Thus, the cube root of 8000 is:
\[ \sqrt[3]{8000} = 20 \]

---