Find the cube root of 85184 by prime factorisation method.

Answer

To find the cube root of 85184 using the prime factorization method, we will follow these steps:

### Step 1: Prime Factorization of 85184
We start by dividing 85184 by the smallest prime number, which is 2.

- 85184 ÷ 2 = 42592
- 42592 ÷ 2 = 21296
- 21296 ÷ 2 = 10648
- 10648 ÷ 2 = 5324
- 5324 ÷ 2 = 2662
- 2662 ÷ 2 = 1331 (Now we can no longer divide by 2)

Next, we divide 1331 by the next smallest prime number, which is 11.

- 1331 ÷ 11 = 121
- 121 ÷ 11 = 11
- 11 ÷ 11 = 1

Now we have completely factored 85184. The prime factorization is:

\[ 85184 = 2^6 \times 11^3 \]

### Step 2: Grouping the Prime Factors
To find the cube root, we need to group the prime factors into triplets (groups of three).

From the factorization:
- \( 2^6 \) can be grouped as \( (2^3) \times (2^3) \)
- \( 11^3 \) is already a complete triplet.

So we can rewrite it as:
\[ 85184 = (2^3) \times (2^3) \times (11^3) \]

### Step 3: Taking the Cube Root
Now we can take the cube root of the grouped factors:

\[ \sqrt[3]{85184} = \sqrt[3]{(2^3) \times (2^3) \times (11^3)} \]

This simplifies to:
\[ \sqrt[3]{(2^3)} \times \sqrt[3]{(2^3)} \times \sqrt[3]{(11^3)} = 2 \times 2 \times 11 \]

Calculating this gives:
\[ 2 \times 2 = 4 \]
\[ 4 \times 11 = 44 \]

### Final Answer
Thus, the cube root of 85184 is:
\[ \sqrt[3]{85184} = 44 \]

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