Find the cube root of 10648 through Prime factorization

Answer

To find the cube root of 10,648 through prime factorization, we can follow these steps:

### Step 1: Prime Factorization of 10,648
We start by dividing 10,648 by the smallest prime number, which is 2.

- 10,648 ÷ 2 = 5,324
- 5,324 ÷ 2 = 2,662
- 2,662 ÷ 2 = 1,331

Now, we cannot divide 1,331 by 2 anymore. Next, we try the next smallest prime number, which is 3.

- 1,331 is not divisible by 3. So, we try the next prime number, which is 5.
- 1,331 is also not divisible by 5. Next, we try 7, 11, and find that:
- 1,331 ÷ 11 = 121
- 121 can be factored as 11 × 11.

So, the complete prime factorization of 10,648 is:
\[ 10,648 = 2^3 \times 11^3 \]

### Step 2: Applying the Cube Root
Now, we need to find the cube root of 10,648:
\[ \sqrt[3]{10,648} = \sqrt[3]{2^3 \times 11^3} \]

### Step 3: Simplifying the Cube Root
Using the property of cube roots, we can separate the factors:
\[ \sqrt[3]{10,648} = \sqrt[3]{2^3} \times \sqrt[3]{11^3} \]

Since the cube root of a number raised to the power of 3 is the base itself:
\[ \sqrt[3]{2^3} = 2 \quad \text{and} \quad \sqrt[3]{11^3} = 11 \]

Thus, we can combine these results:
\[ \sqrt[3]{10,648} = 2 \times 11 = 22 \]

### Final Answer
The cube root of 10,648 is:
\[ \sqrt[3]{10,648} = 22 \]

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