Find the cube root of each of the following numbers by prime factorisation method. 46656

Answer

To find the cube root of 46656 using the prime factorization method, we will follow these steps:

### Step 1: Prime Factorization
First, we need to break down the number 46656 into its prime factors.

- Start dividing 46656 by the smallest prime number, which is 2:
- 46656 ÷ 2 = 23328
- 23328 ÷ 2 = 11664
- 11664 ÷ 2 = 5832
- 5832 ÷ 2 = 2916
- 2916 ÷ 2 = 1458
- 1458 ÷ 2 = 729 (729 is no longer divisible by 2)

Now, we switch to the next smallest prime number, which is 3:
- 729 ÷ 3 = 243
- 243 ÷ 3 = 81
- 81 ÷ 3 = 27
- 27 ÷ 3 = 9
- 9 ÷ 3 = 3
- 3 ÷ 3 = 1

Thus, the complete prime factorization of 46656 is:
\[ 46656 = 2^6 \times 3^6 \]

### Step 2: Grouping the Prime Factors
Next, we group the prime factors in sets of three since we are looking for the cube root.

From the factorization:
\[ 46656 = (2^6) \times (3^6) = (2^3 \times 2^3) \times (3^3 \times 3^3) \]

### Step 3: Taking the Cube Root
Now, we can take the cube root of the grouped factors:
\[ \sqrt[3]{46656} = \sqrt[3]{(2^3 \times 3^3)^2} \]

Since the cube root and cube cancel each other out:
\[ \sqrt[3]{(2^3)} \times \sqrt[3]{(3^3)} = 2 \times 3 \]

### Step 4: Final Calculation
Now we multiply the results:
\[ 2 \times 3 = 6 \]

### Conclusion
Thus, the cube root of 46656 is:
\[ \sqrt[3]{46656} = 36 \]