Evaluate `root3(2744)`.

Options

A. `13`
B. `14`
C. `15`
D. `16`

Answer

To evaluate the cube root of 2744, we can follow these steps:

### Step 1: Identify the number
We need to find the cube root of 2744, which is denoted as \( \sqrt[3]{2744} \).

**Hint:** Remember that the cube root of a number \( x \) is a number \( y \) such that \( y^3 = x \).

### Step 2: Factor the number
We can try to factor 2744 to see if it can be expressed as a product of three identical factors.

**Hint:** Start by checking if 2744 can be divided by small prime numbers like 2, 3, 5, etc.

### Step 3: Check divisibility by 2
Since 2744 is even, we can divide it by 2:
- \( 2744 \div 2 = 1372 \)
- \( 1372 \div 2 = 686 \)
- \( 686 \div 2 = 343 \)

So, we have \( 2744 = 2^3 \times 343 \).

**Hint:** Keep dividing by 2 until you can no longer do so, then check the next smallest prime.

### Step 4: Factor 343
Now we need to factor 343. We can check if it is a perfect cube:
- \( 7 \times 7 = 49 \)
- \( 7 \times 49 = 343 \)

Thus, \( 343 = 7^3 \).

**Hint:** If you recognize a number as a perfect cube, it can simplify your calculations.

### Step 5: Combine the factors
Now we can express 2744 as:
\[ 2744 = 2^3 \times 7^3 \]

**Hint:** Look for patterns in the factors that can help you simplify the expression.

### Step 6: Apply the cube root
Now we can take the cube root of both sides:
\[ \sqrt[3]{2744} = \sqrt[3]{2^3 \times 7^3} \]

Using the property of cube roots, we can separate the factors:
\[ \sqrt[3]{2^3} \times \sqrt[3]{7^3} = 2 \times 7 \]

**Hint:** Remember that \( \sqrt[3]{a^3} = a \).

### Step 7: Calculate the final answer
Now we can calculate:
\[ 2 \times 7 = 14 \]

So, the cube root of 2744 is:
\[ \sqrt[3]{2744} = 14 \]

### Final Answer:
The cube root of 2744 is **14**.

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