Find the values of `root(3)(512)+root(3)(343)+root(3)(729)`

Options

A. 26
B. 24
C. 56
D. 48

Answer

To solve the problem \( \sqrt[3]{512} + \sqrt[3]{343} + \sqrt[3]{729} \), we will break down each term step by step.

### Step 1: Calculate \( \sqrt[3]{512} \)

First, we need to find the cube root of 512.

- We can factor 512:
\[ 512 = 2^9 \quad (\text{since } 512 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2) \]
- Now, we can express it in terms of cube:
\[ 512 = (2^3)^3 \cdot 2^0 = 8^3 \]
- Therefore, the cube root is:
\[ \sqrt[3]{512} = 8 \]

### Step 2: Calculate \( \sqrt[3]{343} \)

Next, we find the cube root of 343.

- We can factor 343:
\[ 343 = 7^3 \]
- Therefore, the cube root is:
\[ \sqrt[3]{343} = 7 \]

### Step 3: Calculate \( \sqrt[3]{729} \)

Now, we find the cube root of 729.

- We can factor 729:
\[ 729 = 9^3 = (3^2)^3 = 3^6 \]
- Therefore, the cube root is:
\[ \sqrt[3]{729} = 9 \]

### Step 4: Combine the results

Now we can combine all the results:
\[ \sqrt[3]{512} + \sqrt[3]{343} + \sqrt[3]{729} = 8 + 7 + 9 \]

### Step 5: Calculate the final sum

Adding these values together:
\[ 8 + 7 + 9 = 24 \]

### Final Answer

Thus, the value of \( \sqrt[3]{512} + \sqrt[3]{343} + \sqrt[3]{729} \) is \( \boxed{24} \).

---