The value of is `root(3)(85184)`

Options

A. 22
B. 33
C. 44
D. 55

Answer

To find the value of \( \sqrt[3]{85184} \), we will follow these steps:

### Step 1: Factorize 85184
We start by factorizing the number 85184. We can check for divisibility by small prime numbers.

- First, we can divide by 2:
\[ 85184 \div 2 = 42592 \]
\[ 42592 \div 2 = 21296 \]
\[ 21296 \div 2 = 10648 \]
\[ 10648 \div 2 = 5324 \]
\[ 5324 \div 2 = 2662 \]
\[ 2662 \div 2 = 1331 \]

Now we have \( 85184 = 2^6 \times 1331 \).

### Step 2: Factor 1331
Next, we need to factor 1331. We can check if it is a perfect cube:
\[ 1331 = 11^3 \]

### Step 3: Combine the factors
Now we can express 85184 in terms of its prime factors:
\[ 85184 = 2^6 \times 11^3 \]

### Step 4: Apply the cube root
Now we apply the cube root to the factored form:
\[ \sqrt[3]{85184} = \sqrt[3]{2^6 \times 11^3} \]

Using the property of cube roots:
\[ \sqrt[3]{a \times b} = \sqrt[3]{a} \times \sqrt[3]{b} \]
we get:
\[ \sqrt[3]{85184} = \sqrt[3]{2^6} \times \sqrt[3]{11^3} \]

### Step 5: Simplify the cube roots
Now we simplify:
\[ \sqrt[3]{2^6} = 2^{6/3} = 2^2 = 4 \]
\[ \sqrt[3]{11^3} = 11^{3/3} = 11 \]

### Step 6: Multiply the results
Now we multiply the results:
\[ \sqrt[3]{85184} = 4 \times 11 = 44 \]

Thus, the value of \( \sqrt[3]{85184} \) is \( \boxed{44} \).

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