Find the value of `root(3)(2744)+7`.

Options

A. 14
B. 21
C. 15
D. 8

Answer

To solve the problem of finding the value of \( \sqrt[3]{2744} + 7 \), we will follow these steps:

### Step 1: Find the cube root of 2744
To find \( \sqrt[3]{2744} \), we need to factor 2744 into its prime factors.

### Step 2: Factor 2744
We can start dividing 2744 by the smallest prime number, which is 2:
- \( 2744 \div 2 = 1372 \)
- \( 1372 \div 2 = 686 \)
- \( 686 \div 2 = 343 \)

Now, 343 is not divisible by 2, so we move to the next prime number, which is 7:
- \( 343 \div 7 = 49 \)
- \( 49 \div 7 = 7 \)
- \( 7 \div 7 = 1 \)

So, the prime factorization of 2744 is:
\[ 2744 = 2^3 \times 7^3 \]

### Step 3: Calculate the cube root
Now, we can find the cube root:
\[ \sqrt[3]{2744} = \sqrt[3]{2^3 \times 7^3} \]
Using the property of cube roots, we can separate the factors:
\[ \sqrt[3]{2^3} \times \sqrt[3]{7^3} = 2 \times 7 = 14 \]

### Step 4: Add 7 to the cube root
Now we add 7 to the cube root we found:
\[ \sqrt[3]{2744} + 7 = 14 + 7 = 21 \]

### Final Answer
Thus, the value of \( \sqrt[3]{2744} + 7 \) is \( 21 \).

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