Find the cube root of 8000.

Options

A. `30`
B. `20`
C. `40`
D. `10`

Explanation

To find the cube root of 8000 using the prime factorization method, we will follow these steps:

### Step 1: Prime Factorization of 8000
We start by dividing 8000 by the smallest prime number, which is 2.

- 8000 ÷ 2 = 4000
- 4000 ÷ 2 = 2000
- 2000 ÷ 2 = 1000
- 1000 ÷ 2 = 500
- 500 ÷ 2 = 250
- 250 ÷ 2 = 125
- 125 ÷ 5 = 25
- 25 ÷ 5 = 5
- 5 ÷ 5 = 1

Now, we can express 8000 as a product of its prime factors:
\[ 8000 = 2^6 \times 5^3 \]

### Step 2: Grouping the Prime Factors
To find the cube root, we need to group the prime factors into triplets (groups of three):
- From \(2^6\), we can form 2 triplets of 2's: \( (2^3) \times (2^3) \)
- From \(5^3\), we can form 1 triplet of 5's: \( (5^3) \)

### Step 3: Taking the Cube Root
Now we can take the cube root of each group:
- The cube root of \(2^6\) is \(2^{6/3} = 2^2 = 4\)
- The cube root of \(5^3\) is \(5^{3/3} = 5^1 = 5\)

### Step 4: Multiply the Results
Now we multiply the results of the cube roots:
\[ \text{Cube root of } 8000 = 4 \times 5 = 20 \]

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

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